WEBVTT 00:00:00.000 --> 00:00:02.395 align:middle line:90% [SQUEAKING] 00:00:02.395 --> 00:00:04.311 align:middle line:90% [RUSTLING] 00:00:04.311 --> 00:00:10.470 align:middle line:90% [CLICKING] 00:00:10.470 --> 00:00:12.115 align:middle line:84% ERIK DEMAINE: Welcome back to 6.1200. 00:00:12.115 --> 00:00:15.080 align:middle line:84% I hope you all had a good spring break. 00:00:15.080 --> 00:00:20.153 align:middle line:84% Today, this week, we continue our segment on graph theory. 00:00:20.153 --> 00:00:21.320 align:middle line:90% I'm going to do more graphs. 00:00:21.320 --> 00:00:23.310 align:middle line:90% Today is undirected graphs. 00:00:23.310 --> 00:00:26.000 align:middle line:84% And next class will be directed graphs. 00:00:26.000 --> 00:00:30.020 align:middle line:84% But remember, an undirected graph, or the full mouthful 00:00:30.020 --> 00:00:32.270 align:middle line:84% is simple undirected graph is what we defined 00:00:32.270 --> 00:00:34.160 align:middle line:90% a couple of lectures ago. 00:00:34.160 --> 00:00:37.890 align:middle line:84% We have a set of vertices, V. We have a set of edges, 00:00:37.890 --> 00:00:40.850 align:middle line:84% E. Vertices can be anything, but the edges 00:00:40.850 --> 00:00:42.900 align:middle line:90% have to be pairs of vertices. 00:00:42.900 --> 00:00:45.480 align:middle line:84% And those vertices have to be distinct. 00:00:45.480 --> 00:00:48.420 align:middle line:84% So the set of edges is always a subset of this. 00:00:48.420 --> 00:00:50.810 align:middle line:84% This would be the maximum possible set of edges. 00:00:50.810 --> 00:00:55.470 align:middle line:84% You have two-item set u, v for all vertices u and v, 00:00:55.470 --> 00:00:58.020 align:middle line:90% where u does not equal v. 00:00:58.020 --> 00:01:01.240 align:middle line:84% So today I'm just going to remove-- 00:01:01.240 --> 00:01:03.350 align:middle line:84% omit these adjectives and just say graph. 00:01:03.350 --> 00:01:05.019 align:middle line:84% But whenever I say graph today, I really 00:01:05.019 --> 00:01:07.090 align:middle line:90% mean simple undirected graph. 00:01:07.090 --> 00:01:10.220 align:middle line:84% And we're going to walk around in graphs today. 00:01:10.220 --> 00:01:14.530 align:middle line:84% So let's define a notion of walk. 00:01:14.530 --> 00:01:19.310 align:middle line:84% This is going to be similar to an execution in state machines. 00:01:19.310 --> 00:01:21.880 align:middle line:84% Next lecture will be even more like that. 00:01:21.880 --> 00:01:27.280 align:middle line:84% But for today a walk is a sequence of vertices. 00:01:27.280 --> 00:01:30.320 align:middle line:84% I'm going to write that sequence with arrows, 00:01:30.320 --> 00:01:34.270 align:middle line:90% but it's really just a list. 00:01:34.270 --> 00:01:38.230 align:middle line:84% It's a finite sequence, let's say, 00:01:38.230 --> 00:01:49.510 align:middle line:84% where each consecutive vi to vi plus 1 is an edge in the graph. 00:01:49.510 --> 00:01:53.800 align:middle line:84% So this is like a valid transition in the state machine 00:01:53.800 --> 00:01:55.480 align:middle line:90% world. 00:01:55.480 --> 00:02:02.480 align:middle line:84% So this is like i equals 0, 1 up to k minus 1 00:02:02.480 --> 00:02:05.000 align:middle line:90% is when that needs to apply. 00:02:05.000 --> 00:02:06.980 align:middle line:90% So if you have-- 00:02:06.980 --> 00:02:08.560 align:middle line:90% let me draw a graph here. 00:02:08.560 --> 00:02:14.960 align:middle line:90% 00:02:14.960 --> 00:02:18.380 align:middle line:84% It's one of the first graphs I ever drew in my life. 00:02:18.380 --> 00:02:22.700 align:middle line:84% I still remember it for reasons we will talk about in a moment. 00:02:22.700 --> 00:02:25.970 align:middle line:84% Here's a graph with 5 vertices, a bunch of edges. 00:02:25.970 --> 00:02:27.790 align:middle line:90% And let me get a color. 00:02:27.790 --> 00:02:32.510 align:middle line:90% 00:02:32.510 --> 00:02:41.930 align:middle line:84% So for example, this is a path, which is-- 00:02:41.930 --> 00:02:52.550 align:middle line:84% sorry-- is a walk, which I would write a to b to d to e 00:02:52.550 --> 00:02:56.890 align:middle line:90% to b back to d up to c. 00:02:56.890 --> 00:02:58.250 align:middle line:90% That's writing it with commas. 00:02:58.250 --> 00:02:59.667 align:middle line:84% You can also write it with arrows. 00:02:59.667 --> 00:03:01.340 align:middle line:90% It's the same thing. 00:03:01.340 --> 00:03:04.030 align:middle line:90% So that's an example of a walk. 00:03:04.030 --> 00:03:07.250 align:middle line:84% And for some terminology-- it would be helpful-- 00:03:07.250 --> 00:03:13.880 align:middle line:84% we're going to call this a walk from v0 to vk. 00:03:13.880 --> 00:03:16.360 align:middle line:84% V0 is just the first vertex in the sequence. vk 00:03:16.360 --> 00:03:18.080 align:middle line:90% is the last sequence. 00:03:18.080 --> 00:03:21.100 align:middle line:84% But I'm defining what from and to means. 00:03:21.100 --> 00:03:26.230 align:middle line:84% And we'll define the length of this path to equal k. 00:03:26.230 --> 00:03:31.190 align:middle line:84% k is the number of edges in this sequence, 00:03:31.190 --> 00:03:32.470 align:middle line:90% not the number of vertices. 00:03:32.470 --> 00:03:35.360 align:middle line:84% The number of edges is one fewer than the number of vertices. 00:03:35.360 --> 00:03:39.350 align:middle line:84% So in this example, we have 1, 2, 3, 4, 5, 6 edges. 00:03:39.350 --> 00:03:43.550 align:middle line:90% So this is a length 6 walk. 00:03:43.550 --> 00:03:50.230 align:middle line:84% In particular, we allow length to equal 0. 00:03:50.230 --> 00:03:52.310 align:middle line:84% What does a length zero walk look like? 00:03:52.310 --> 00:03:54.680 align:middle line:84% Well, it's a vertex, and that's it. 00:03:54.680 --> 00:03:55.800 align:middle line:90% You just stay there. 00:03:55.800 --> 00:03:59.100 align:middle line:84% So c by itself is a zero length walk. 00:03:59.100 --> 00:04:03.000 align:middle line:84% e by itself is a zero length walk. 00:04:03.000 --> 00:04:06.950 align:middle line:90% Not so exciting yet. 00:04:06.950 --> 00:04:09.685 align:middle line:84% Let's define a related notion, which is path. 00:04:09.685 --> 00:04:12.830 align:middle line:90% 00:04:12.830 --> 00:04:20.930 align:middle line:84% And this is a special kind of walk with no repeated vertices 00:04:20.930 --> 00:04:21.595 align:middle line:90% or edges. 00:04:21.595 --> 00:04:30.020 align:middle line:90% 00:04:30.020 --> 00:04:32.700 align:middle line:84% So in particular, not this path-- 00:04:32.700 --> 00:04:37.415 align:middle line:84% this walk is not a path, because it visits some vertices 00:04:37.415 --> 00:04:38.040 align:middle line:90% multiple times. 00:04:38.040 --> 00:04:38.953 align:middle line:90% It visits b twice. 00:04:38.953 --> 00:04:40.370 align:middle line:84% You can see that in this sequence. 00:04:40.370 --> 00:04:42.620 align:middle line:84% It also visits some edges multiple times. b 00:04:42.620 --> 00:04:45.020 align:middle line:90% to d is repeated. 00:04:45.020 --> 00:05:01.120 align:middle line:84% So if I redraw this graph, an example 00:05:01.120 --> 00:05:05.660 align:middle line:84% of a path between the same two vertices would be this one. 00:05:05.660 --> 00:05:08.350 align:middle line:84% This is still starting at a and ending at c. 00:05:08.350 --> 00:05:12.093 align:middle line:84% But now it's shorter, and it doesn't repeat any vertices. 00:05:12.093 --> 00:05:14.920 align:middle line:90% 00:05:14.920 --> 00:05:20.235 align:middle line:84% In particular, with walks, if you don't repeat a vertex, 00:05:20.235 --> 00:05:21.860 align:middle line:84% you probably also don't repeat an edge. 00:05:21.860 --> 00:05:24.160 align:middle line:84% But I think it's helpful here to highlight we really don't want 00:05:24.160 --> 00:05:25.370 align:middle line:90% to repeat vertices or edges. 00:05:25.370 --> 00:05:26.890 align:middle line:84% This will get a little more subtle 00:05:26.890 --> 00:05:31.150 align:middle line:84% when we talk about closed walks in a little bit. 00:05:31.150 --> 00:05:33.740 align:middle line:84% So why might we care about walking in a graph? 00:05:33.740 --> 00:05:35.510 align:middle line:90% Lots of practical motivations. 00:05:35.510 --> 00:05:37.420 align:middle line:84% Every time you, I don't know, look up 00:05:37.420 --> 00:05:41.680 align:middle line:84% something in Google Maps, you're finding a path from a to b. 00:05:41.680 --> 00:05:45.340 align:middle line:84% Maybe you want the shortest path, or the least energy path, 00:05:45.340 --> 00:05:47.290 align:middle line:90% or so on. 00:05:47.290 --> 00:05:48.740 align:middle line:90% So we care about paths a lot. 00:05:48.740 --> 00:05:51.140 align:middle line:84% When we're finding routes in transportation networks, 00:05:51.140 --> 00:05:53.780 align:middle line:84% we care about them when we're finding routes on the internet 00:05:53.780 --> 00:05:55.670 align:middle line:90% or on a computer network. 00:05:55.670 --> 00:05:58.070 align:middle line:84% In both those cases, the vertices are places you can be, 00:05:58.070 --> 00:06:01.590 align:middle line:84% and the edges are the routes, like the streets you can take, 00:06:01.590 --> 00:06:03.020 align:middle line:90% for example. 00:06:03.020 --> 00:06:05.300 align:middle line:84% In a social network, if the vertices are people 00:06:05.300 --> 00:06:07.190 align:middle line:84% and the edges are friendships, then 00:06:07.190 --> 00:06:10.020 align:middle line:84% you might care about degrees of separation between people, 00:06:10.020 --> 00:06:13.790 align:middle line:84% like a famous experiment where they asked people 00:06:13.790 --> 00:06:16.520 align:middle line:84% to send physical letters to people 00:06:16.520 --> 00:06:17.940 align:middle line:90% they knew on a first name basis. 00:06:17.940 --> 00:06:19.140 align:middle line:90% This is like in the '60s-- 00:06:19.140 --> 00:06:20.600 align:middle line:90% Milgram. 00:06:20.600 --> 00:06:24.200 align:middle line:84% And they found that the distance between two people 00:06:24.200 --> 00:06:25.230 align:middle line:90% is only about six. 00:06:25.230 --> 00:06:26.490 align:middle line:90% These days it's about five. 00:06:26.490 --> 00:06:29.360 align:middle line:84% We're getting more connected, despite more people. 00:06:29.360 --> 00:06:32.010 align:middle line:84% Anyway, those are some motivations. 00:06:32.010 --> 00:06:35.810 align:middle line:84% Also, in things like the 8 puzzle 00:06:35.810 --> 00:06:38.220 align:middle line:84% that we talked about some time ago, 00:06:38.220 --> 00:06:40.670 align:middle line:84% you can think of each of these configurations 00:06:40.670 --> 00:06:43.050 align:middle line:90% as vertices in a graph. 00:06:43.050 --> 00:06:45.330 align:middle line:90% And the edges are transitions. 00:06:45.330 --> 00:06:48.990 align:middle line:84% And because every time I make a move, 00:06:48.990 --> 00:06:50.640 align:middle line:90% I can also unmake that move. 00:06:50.640 --> 00:06:52.340 align:middle line:90% This is an undirected graph. 00:06:52.340 --> 00:06:54.470 align:middle line:84% Everywhere I can go, I can immediately go back. 00:06:54.470 --> 00:06:58.560 align:middle line:84% And you might care about, what does this graph look like? 00:06:58.560 --> 00:06:59.060 align:middle line:90% It's big. 00:06:59.060 --> 00:07:00.980 align:middle line:90% It's got like 100,000 vertices. 00:07:00.980 --> 00:07:02.420 align:middle line:90% So I can't draw it here. 00:07:02.420 --> 00:07:05.630 align:middle line:84% But you can explore it by moving around. 00:07:05.630 --> 00:07:07.540 align:middle line:90% Can you get everywhere? 00:07:07.540 --> 00:07:08.750 align:middle line:90% That's a good question. 00:07:08.750 --> 00:07:10.600 align:middle line:90% And that's the next topic. 00:07:10.600 --> 00:07:15.370 align:middle line:84% Connectivity is what this lecture is mostly about. 00:07:15.370 --> 00:07:19.886 align:middle line:84% So let's talk about connectivity. 00:07:19.886 --> 00:07:24.430 align:middle line:90% 00:07:24.430 --> 00:07:28.390 align:middle line:84% So we'll call two vertices connected 00:07:28.390 --> 00:07:30.165 align:middle line:90% if there's a walk between them. 00:07:30.165 --> 00:07:44.440 align:middle line:90% 00:07:44.440 --> 00:07:46.840 align:middle line:84% I'd like to say there's a walk between them, which 00:07:46.840 --> 00:07:49.180 align:middle line:84% is not caring about which one you start at 00:07:49.180 --> 00:07:50.960 align:middle line:90% and which you end at. 00:07:50.960 --> 00:07:54.860 align:middle line:84% But the way we define walk is it starts at one place 00:07:54.860 --> 00:07:56.460 align:middle line:90% and it goes to the other. 00:07:56.460 --> 00:07:59.270 align:middle line:84% Of course, if v0 to vk is a walk, then 00:07:59.270 --> 00:08:02.420 align:middle line:84% also vk backwards to v0 is also a walk. 00:08:02.420 --> 00:08:05.630 align:middle line:84% If I flip this sequence, if I reverse it, 00:08:05.630 --> 00:08:09.072 align:middle line:84% it's still a walk, because edges are sets. 00:08:09.072 --> 00:08:11.280 align:middle line:84% They don't care about which order you follow them in. 00:08:11.280 --> 00:08:12.900 align:middle line:84% So if you can go from vi to vi plus 1, 00:08:12.900 --> 00:08:17.240 align:middle line:84% you can go from vi plus 1 to vi in an undirected graph. 00:08:17.240 --> 00:08:25.992 align:middle line:84% So this is equivalent to there existing a walk from v to u. 00:08:25.992 --> 00:08:27.950 align:middle line:84% It doesn't matter which one you put on the left 00:08:27.950 --> 00:08:29.242 align:middle line:90% and which you put on the right. 00:08:29.242 --> 00:08:30.830 align:middle line:90% These are the same thing. 00:08:30.830 --> 00:08:32.125 align:middle line:90% Technically, this is a theorem. 00:08:32.125 --> 00:08:33.000 align:middle line:90% You have to prove it. 00:08:33.000 --> 00:08:35.320 align:middle line:84% And the proof is just reverse the walk. 00:08:35.320 --> 00:08:42.020 align:middle line:90% 00:08:42.020 --> 00:08:44.850 align:middle line:84% If I have a walk from u to v, I can reverse it. 00:08:44.850 --> 00:08:46.110 align:middle line:90% And I get a walk from v to u. 00:08:46.110 --> 00:08:48.652 align:middle line:84% If I have a walk from v to u, I can reverse it and get a walk 00:08:48.652 --> 00:08:50.170 align:middle line:84% from u to v. So in either direction, 00:08:50.170 --> 00:08:51.560 align:middle line:90% these two things are equivalent. 00:08:51.560 --> 00:08:53.930 align:middle line:84% And so we just say there's a walk between u and v, 00:08:53.930 --> 00:08:56.230 align:middle line:84% or we say u and v are connected, or v and u are 00:08:56.230 --> 00:08:58.930 align:middle line:90% connected-- all the same thing. 00:08:58.930 --> 00:09:02.600 align:middle line:84% There are some other equivalent properties to this. 00:09:02.600 --> 00:09:05.590 align:middle line:90% So I want to write some more. 00:09:05.590 --> 00:09:10.515 align:middle line:84% The other big one is there exists a path from v to u. 00:09:10.515 --> 00:09:14.320 align:middle line:90% 00:09:14.320 --> 00:09:17.300 align:middle line:84% This is perhaps a good moment to mention-- 00:09:17.300 --> 00:09:19.420 align:middle line:84% so we're using the words walk and path 00:09:19.420 --> 00:09:23.240 align:middle line:84% here in one consistent way throughout this class. 00:09:23.240 --> 00:09:25.660 align:middle line:84% But there are other classes, such as the very next class, 00:09:25.660 --> 00:09:29.630 align:middle line:84% 6.121, where this is called path, 00:09:29.630 --> 00:09:31.130 align:middle line:90% and this is called simple path. 00:09:31.130 --> 00:09:32.330 align:middle line:90% So just a warning-- 00:09:32.330 --> 00:09:33.760 align:middle line:84% there's two sets of terminologies 00:09:33.760 --> 00:09:36.340 align:middle line:84% for whether a walk is allowed to-- 00:09:36.340 --> 00:09:37.870 align:middle line:90% walks can always repeat. 00:09:37.870 --> 00:09:40.910 align:middle line:84% Vertices, paths, sometimes they can. 00:09:40.910 --> 00:09:41.870 align:middle line:90% Sometimes they can't. 00:09:41.870 --> 00:09:43.550 align:middle line:90% And in this class, they cannot. 00:09:43.550 --> 00:09:46.030 align:middle line:84% So we're going to use walk to mean the general thing, 00:09:46.030 --> 00:09:48.155 align:middle line:84% and path to mean the specific thing where you don't 00:09:48.155 --> 00:09:51.100 align:middle line:90% repeat any vertices or edges. 00:09:51.100 --> 00:09:54.900 align:middle line:84% So I claim there's a walk from u to v if and only 00:09:54.900 --> 00:09:58.055 align:middle line:84% if there's a path from u to v. Let's prove this. 00:09:58.055 --> 00:10:03.390 align:middle line:90% 00:10:03.390 --> 00:10:06.610 align:middle line:84% So remember, how do we prove that two things are equivalent? 00:10:06.610 --> 00:10:08.040 align:middle line:84% How do we prove an if and only if? 00:10:08.040 --> 00:10:09.310 align:middle line:90% We do it in two parts. 00:10:09.310 --> 00:10:11.770 align:middle line:84% We prove left to right and right to left. 00:10:11.770 --> 00:10:14.670 align:middle line:84% So I'm going to do right to left first, 00:10:14.670 --> 00:10:16.630 align:middle line:90% because this is the easy one. 00:10:16.630 --> 00:10:21.820 align:middle line:84% Why is it, I ask you, if I have a path from u to v, 00:10:21.820 --> 00:10:25.050 align:middle line:84% is there necessarily a walk from u to v? 00:10:25.050 --> 00:10:27.660 align:middle line:90% Any suggestions? 00:10:27.660 --> 00:10:28.238 align:middle line:90% Yeah. 00:10:28.238 --> 00:10:29.988 align:middle line:84% AUDIENCE: By definition, a path is a walk. 00:10:29.988 --> 00:10:31.690 align:middle line:90% ERIK DEMAINE: A path is a walk. 00:10:31.690 --> 00:10:33.370 align:middle line:84% A path is a special kind of walk. 00:10:33.370 --> 00:10:35.730 align:middle line:84% So if I have a path, then definitely I have a walk, 00:10:35.730 --> 00:10:37.240 align:middle line:90% because that path is the walk. 00:10:37.240 --> 00:10:44.040 align:middle line:84% So I would usually write a path is a kind of walk. 00:10:44.040 --> 00:10:46.450 align:middle line:90% So that's the easy direction. 00:10:46.450 --> 00:10:50.150 align:middle line:84% The more impressive thing is that if there's a walk, 00:10:50.150 --> 00:10:52.760 align:middle line:84% then there's actually-- not only is there a walk, 00:10:52.760 --> 00:10:56.030 align:middle line:84% but there's a walk that doesn't repeat any vertices or edges. 00:10:56.030 --> 00:10:57.820 align:middle line:90% This is trickier. 00:10:57.820 --> 00:11:04.270 align:middle line:84% And I'm going to tell you how to find it, 00:11:04.270 --> 00:11:06.495 align:middle line:90% because it needs a nifty idea. 00:11:06.495 --> 00:11:09.160 align:middle line:90% 00:11:09.160 --> 00:11:11.510 align:middle line:84% But first, we write the template, 00:11:11.510 --> 00:11:15.680 align:middle line:84% which is we want to prove that if there's a walk from u to v, 00:11:15.680 --> 00:11:16.550 align:middle line:90% then there's a path. 00:11:16.550 --> 00:11:21.250 align:middle line:84% So let's assume there's a walk from u to v. And let's give it-- 00:11:21.250 --> 00:11:23.230 align:middle line:84% I'm not going to give it a name yet. 00:11:23.230 --> 00:11:25.010 align:middle line:90% So there exists a walk. 00:11:25.010 --> 00:11:26.240 align:middle line:90% There might be many walks. 00:11:26.240 --> 00:11:29.180 align:middle line:84% I want to try to choose not necessarily a particular one, 00:11:29.180 --> 00:11:32.950 align:middle line:84% but a more restricted type of walk that's possible, because I 00:11:32.950 --> 00:11:35.140 align:middle line:90% know that these at least exist. 00:11:35.140 --> 00:11:37.725 align:middle line:84% So I want to take a shortest walk-- 00:11:37.725 --> 00:11:46.350 align:middle line:90% 00:11:46.350 --> 00:11:51.090 align:middle line:84% shortest in terms of length, so minimum possible length. 00:11:51.090 --> 00:11:53.670 align:middle line:84% Among all walks, I want one of them 00:11:53.670 --> 00:11:56.770 align:middle line:84% that minimizes the length of the walk. 00:11:56.770 --> 00:11:58.150 align:middle line:90% We know there exists some walk. 00:11:58.150 --> 00:12:01.110 align:middle line:84% So there is some solution here because you can't 00:12:01.110 --> 00:12:02.980 align:middle line:90% go below a walk of length zero. 00:12:02.980 --> 00:12:05.610 align:middle line:90% There's some shortest. 00:12:05.610 --> 00:12:07.870 align:middle line:84% And this one I'm going to give a name. 00:12:07.870 --> 00:12:11.350 align:middle line:84% Let's say it starts at u, which is also called v0. 00:12:11.350 --> 00:12:14.040 align:middle line:84% Then it goes to v1, then it goes to v2, 00:12:14.040 --> 00:12:19.980 align:middle line:84% and so on, until it reaches vk, which equals v. So it's a walk 00:12:19.980 --> 00:12:25.930 align:middle line:84% from u to v, using some number of edges in between. 00:12:25.930 --> 00:12:27.470 align:middle line:90% But now I know it's shortest. 00:12:27.470 --> 00:12:31.740 align:middle line:90% 00:12:31.740 --> 00:12:42.330 align:middle line:84% So I claim that this is a path from u 00:12:42.330 --> 00:12:44.980 align:middle line:84% to v. I claim that there are no repeated vertices 00:12:44.980 --> 00:12:47.560 align:middle line:90% or edges in this sequence. 00:12:47.560 --> 00:12:51.290 align:middle line:84% In particular, I claim that there are no repeated vertices. 00:12:51.290 --> 00:13:06.400 align:middle line:84% So let's prove the claim, so maybe as a picture first. 00:13:06.400 --> 00:13:12.280 align:middle line:84% So suppose that-- let's do it by contradiction. 00:13:12.280 --> 00:13:17.930 align:middle line:84% Let's suppose that the path does repeat a vertex. 00:13:17.930 --> 00:13:21.550 align:middle line:84% So it starts at u, and then it goes-- 00:13:21.550 --> 00:13:27.640 align:middle line:84% which is v0, then it goes to v1 and so on. 00:13:27.640 --> 00:13:33.640 align:middle line:84% Maybe at some point after going for a while, 00:13:33.640 --> 00:13:36.450 align:middle line:84% it comes back to a vertex it's been to 00:13:36.450 --> 00:13:40.490 align:middle line:84% before proceeding to vk over here, 00:13:40.490 --> 00:13:46.290 align:middle line:84% which is v. This is what a walk looks like that repeats at least 00:13:46.290 --> 00:13:47.440 align:middle line:90% one vertex. 00:13:47.440 --> 00:13:48.940 align:middle line:84% I claim this is the general picture, 00:13:48.940 --> 00:13:51.810 align:middle line:84% but you can write it down in words. 00:13:51.810 --> 00:13:55.750 align:middle line:84% Then the idea is, well, if the walk did this-- 00:13:55.750 --> 00:14:00.210 align:middle line:84% maybe I should draw this as a continuous path in red. 00:14:00.210 --> 00:14:04.810 align:middle line:84% If the walk did this, where it repeats this vertex twice, 00:14:04.810 --> 00:14:06.870 align:middle line:90% why not just do this? 00:14:06.870 --> 00:14:08.730 align:middle line:90% That's a shorter path. 00:14:08.730 --> 00:14:11.620 align:middle line:84% And that contradicts that we started with the shortest path. 00:14:11.620 --> 00:14:13.900 align:middle line:84% If this path is strictly shorter than this one, 00:14:13.900 --> 00:14:15.780 align:middle line:90% that's a contradiction. 00:14:15.780 --> 00:14:19.050 align:middle line:84% Write that down a little more carefully. 00:14:19.050 --> 00:14:24.690 align:middle line:84% So for contradiction, let's say that vi equals 00:14:24.690 --> 00:14:28.830 align:middle line:90% vj for some i less than j. 00:14:28.830 --> 00:14:35.370 align:middle line:84% So this would be both vi and vj are the same vertex 00:14:35.370 --> 00:14:36.820 align:middle line:90% in this sequence. 00:14:36.820 --> 00:14:38.170 align:middle line:90% The vertex gets repeated. 00:14:38.170 --> 00:14:39.610 align:middle line:90% That's the general situation. 00:14:39.610 --> 00:14:42.250 align:middle line:84% I can assume i is less than j, just by relabeling i and j, 00:14:42.250 --> 00:14:44.030 align:middle line:90% swapping i and j if I need to. 00:14:44.030 --> 00:14:51.730 align:middle line:84% And now the idea is that v0 up to vi, which equals vj, 00:14:51.730 --> 00:14:56.980 align:middle line:84% and then going directly to vj plus 1, vj plus 2, 00:14:56.980 --> 00:15:07.630 align:middle line:84% and so on, up to vk is a shorter walk between the same two 00:15:07.630 --> 00:15:12.410 align:middle line:84% vertices, still from u to v. And so that is a contradiction. 00:15:12.410 --> 00:15:17.590 align:middle line:90% 00:15:17.590 --> 00:15:20.030 align:middle line:84% And so that ends the proof of the claim. 00:15:20.030 --> 00:15:24.250 align:middle line:84% And that ends the proof of this claim, that if there's a walk, 00:15:24.250 --> 00:15:25.460 align:middle line:90% I can turn it into a path. 00:15:25.460 --> 00:15:27.650 align:middle line:90% I take the shortest walk. 00:15:27.650 --> 00:15:32.650 align:middle line:84% And it turns out that must be a path-- proof by contradiction. 00:15:32.650 --> 00:15:34.550 align:middle line:90% So a bunch of work there. 00:15:34.550 --> 00:15:37.340 align:middle line:84% But now, for connectivity's sake, 00:15:37.340 --> 00:15:38.890 align:middle line:84% we can use walk when it's convenient. 00:15:38.890 --> 00:15:41.820 align:middle line:84% We can use path when it's convenient. 00:15:41.820 --> 00:15:44.620 align:middle line:84% For connectivity, these two notions are interchangeable. 00:15:44.620 --> 00:15:47.650 align:middle line:90% We can use whatever is nicest. 00:15:47.650 --> 00:15:51.510 align:middle line:84% Let me mention a couple other fun properties of connectivity 00:15:51.510 --> 00:15:53.550 align:middle line:90% over here. 00:15:53.550 --> 00:15:57.060 align:middle line:84% So this one, walk from u to v versus v to u 00:15:57.060 --> 00:15:58.633 align:middle line:90% is a property we call symmetry. 00:15:58.633 --> 00:16:01.980 align:middle line:90% 00:16:01.980 --> 00:16:04.740 align:middle line:84% And there are two other properties 00:16:04.740 --> 00:16:06.790 align:middle line:90% that are nice to know. 00:16:06.790 --> 00:16:14.380 align:middle line:84% One is called reflexivity, or this is the reflexive property. 00:16:14.380 --> 00:16:18.150 align:middle line:84% Every vertex is connected to itself. 00:16:18.150 --> 00:16:19.470 align:middle line:90% Why? 00:16:19.470 --> 00:16:23.730 align:middle line:84% Because we allowed zero length walks and paths. 00:16:23.730 --> 00:16:27.070 align:middle line:84% Starting at v and doing nothing, that is a valid walk. 00:16:27.070 --> 00:16:29.880 align:middle line:90% So that's a walk from v to v. 00:16:29.880 --> 00:16:34.516 align:middle line:84% And another useful fact is transitivity. 00:16:34.516 --> 00:16:37.070 align:middle line:90% 00:16:37.070 --> 00:16:40.340 align:middle line:84% Connectedness is a transitive relation. 00:16:40.340 --> 00:16:45.770 align:middle line:84% If we have u and v are connected and we 00:16:45.770 --> 00:16:56.400 align:middle line:84% have v and w are connected, then we know u and w are connected. 00:16:56.400 --> 00:16:58.910 align:middle line:90% 00:16:58.910 --> 00:17:00.950 align:middle line:90% Why? 00:17:00.950 --> 00:17:04.170 align:middle line:84% This is a particularly nice proof by picture. 00:17:04.170 --> 00:17:07.170 align:middle line:84% Please don't think that proof by picture is a valid proof method, 00:17:07.170 --> 00:17:10.349 align:middle line:84% but I'll tell you the words in a second. 00:17:10.349 --> 00:17:13.410 align:middle line:84% Suppose we have vertices u, v, and w. 00:17:13.410 --> 00:17:15.710 align:middle line:84% And there's some path from u to v. And there's 00:17:15.710 --> 00:17:17.750 align:middle line:90% some path from v to w. 00:17:17.750 --> 00:17:20.250 align:middle line:84% Well, then there's a path from u to w. 00:17:20.250 --> 00:17:27.113 align:middle line:84% You just concatenate the paths, or concatenate the walks. 00:17:27.113 --> 00:17:30.080 align:middle line:90% 00:17:30.080 --> 00:17:32.760 align:middle line:84% At this point, we could assume we had paths initially, 00:17:32.760 --> 00:17:35.940 align:middle line:84% but when we glue them together, it won't be a path anymore 00:17:35.940 --> 00:17:36.820 align:middle line:90% necessarily. 00:17:36.820 --> 00:17:37.660 align:middle line:90% It might be a walk. 00:17:37.660 --> 00:17:39.577 align:middle line:84% There might be some repetitions here and here. 00:17:39.577 --> 00:17:43.912 align:middle line:84% But for the easy proof, just take two walks. 00:17:43.912 --> 00:17:44.620 align:middle line:90% Concatenate them. 00:17:44.620 --> 00:17:45.460 align:middle line:90% It's still a walk. 00:17:45.460 --> 00:17:48.720 align:middle line:84% Therefore, u and w are connected. 00:17:48.720 --> 00:17:51.160 align:middle line:84% Next week we'll talk about these properties-- symmetry, 00:17:51.160 --> 00:17:53.130 align:middle line:84% reflexivity, and transitivity-- even more, 00:17:53.130 --> 00:17:55.980 align:middle line:84% so this is kind of a foreshadowing of that. 00:17:55.980 --> 00:18:03.010 align:middle line:84% But it lets us define some cool concepts about connectivity. 00:18:03.010 --> 00:18:12.840 align:middle line:84% So first, we're going to call a graph connected 00:18:12.840 --> 00:18:23.550 align:middle line:84% if u and v are connected for all vertices u and v. 00:18:23.550 --> 00:18:26.860 align:middle line:84% So we sometimes call this pairwise connected. 00:18:26.860 --> 00:18:32.100 align:middle line:84% Pairwise is a useful notion when you have something 00:18:32.100 --> 00:18:34.260 align:middle line:84% that only makes sense on pairs of elements, 00:18:34.260 --> 00:18:36.930 align:middle line:84% but you want it to apply to the whole structure, 00:18:36.930 --> 00:18:39.710 align:middle line:84% in this case, the whole graph, not just pairs of vertices. 00:18:39.710 --> 00:18:42.800 align:middle line:84% Then you just put a for all over the two things. 00:18:42.800 --> 00:18:44.990 align:middle line:84% And that would be pairwise connected. 00:18:44.990 --> 00:18:48.290 align:middle line:84% But it really doesn't matter how you-- 00:18:48.290 --> 00:18:51.410 align:middle line:84% connected is kind of a robust notion because 00:18:51.410 --> 00:18:54.050 align:middle line:90% of things like transitivity. 00:18:54.050 --> 00:18:56.270 align:middle line:84% In an undirected graph, it's the same to say 00:18:56.270 --> 00:18:58.320 align:middle line:84% that every pair of vertices is connected, 00:18:58.320 --> 00:19:00.470 align:middle line:84% as it is to say that there's some vertex 00:19:00.470 --> 00:19:03.290 align:middle line:84% that every other vertex is connected 00:19:03.290 --> 00:19:06.330 align:middle line:90% to because of transitivity. 00:19:06.330 --> 00:19:09.030 align:middle line:84% We know if you can get from a to b and from b to c, 00:19:09.030 --> 00:19:12.170 align:middle line:84% then you can also get from a to c. 00:19:12.170 --> 00:19:16.820 align:middle line:84% So this is why connectivity is the right notion. 00:19:16.820 --> 00:19:19.550 align:middle line:84% Even more interesting is a notion 00:19:19.550 --> 00:19:21.256 align:middle line:90% called connected components. 00:19:21.256 --> 00:19:49.200 align:middle line:90% 00:19:49.200 --> 00:19:53.350 align:middle line:84% Maybe let me draw an example before I define this carefully. 00:19:53.350 --> 00:19:55.800 align:middle line:84% So this graph, for example, is connected. 00:19:55.800 --> 00:19:58.090 align:middle line:84% From every vertex you can get to any other. 00:19:58.090 --> 00:19:59.865 align:middle line:90% Let's draw a disconnected graph. 00:19:59.865 --> 00:20:04.270 align:middle line:90% 00:20:04.270 --> 00:20:07.150 align:middle line:84% I should have it over here somewhere. 00:20:07.150 --> 00:20:09.930 align:middle line:90% 00:20:09.930 --> 00:20:13.620 align:middle line:84% Let's say I've got this guy here, 00:20:13.620 --> 00:20:23.470 align:middle line:84% and maybe just a vertex and some edge, a, b, c, d, e, f, g. 00:20:23.470 --> 00:20:25.180 align:middle line:90% So this is one graph. 00:20:25.180 --> 00:20:30.250 align:middle line:90% It has six vertices, four edges. 00:20:30.250 --> 00:20:32.430 align:middle line:84% And it's, obviously, not connected. 00:20:32.430 --> 00:20:34.340 align:middle line:84% There's no walk, for example, from here 00:20:34.340 --> 00:20:40.850 align:middle line:84% to here, or from here to there, b to f, c to e, whatever. 00:20:40.850 --> 00:20:43.940 align:middle line:84% But it's still possible to get from a to c, a to b, a to d. 00:20:43.940 --> 00:20:46.710 align:middle line:84% So a can get to everyone in this chunk. 00:20:46.710 --> 00:20:48.870 align:middle line:84% And that chunk we call a connected component. 00:20:48.870 --> 00:20:51.450 align:middle line:84% So in this graph, there are three connected components. 00:20:51.450 --> 00:20:54.710 align:middle line:84% These are the mutually reachable regions of the graph. 00:20:54.710 --> 00:20:57.380 align:middle line:90% 00:20:57.380 --> 00:21:00.080 align:middle line:84% So to define that precisely, I'm going 00:21:00.080 --> 00:21:04.233 align:middle line:84% to introduce a new term, which is induced subgraph. 00:21:04.233 --> 00:21:20.780 align:middle line:90% 00:21:20.780 --> 00:21:24.050 align:middle line:84% So what does induced subgraph mean? 00:21:24.050 --> 00:21:25.680 align:middle line:90% It's a notion. 00:21:25.680 --> 00:21:27.860 align:middle line:84% Given one graph, it defines a graph 00:21:27.860 --> 00:21:29.640 align:middle line:84% that lives inside the original graph. 00:21:29.640 --> 00:21:31.810 align:middle line:84% It's a subset of the vertices and edges. 00:21:31.810 --> 00:21:34.290 align:middle line:90% That's what subgraph means. 00:21:34.290 --> 00:21:37.750 align:middle line:84% And it's defined by a set of vertices. 00:21:37.750 --> 00:21:40.750 align:middle line:84% So it's the subgraph induced by a particular set of vertices. 00:21:40.750 --> 00:21:45.210 align:middle line:84% In this case, the vertex set is all the vertices 00:21:45.210 --> 00:21:48.340 align:middle line:84% connected to v. This is a nice shorthand for saying, 00:21:48.340 --> 00:21:52.630 align:middle line:84% give me all the u, where u and v are connected. 00:21:52.630 --> 00:21:54.978 align:middle line:90% 00:21:54.978 --> 00:21:57.270 align:middle line:84% And as we said, it doesn't matter whether I say u and v 00:21:57.270 --> 00:22:00.420 align:middle line:90% or v and u-- same thing. 00:22:00.420 --> 00:22:05.118 align:middle line:84% And the edge set is determined by this word induced. 00:22:05.118 --> 00:22:07.410 align:middle line:84% When I say induced, I mean I just specify the vertices, 00:22:07.410 --> 00:22:10.750 align:middle line:84% and then I want all the edges that could possibly make sense. 00:22:10.750 --> 00:22:15.930 align:middle line:84% So in other words, I want to take all of the edges u, 00:22:15.930 --> 00:22:20.610 align:middle line:84% v that were in my original edge set with the property 00:22:20.610 --> 00:22:25.740 align:middle line:84% that both u and v are in my new vertex set. 00:22:25.740 --> 00:22:31.580 align:middle line:84% So for example, in this graph, if I take the subgraph induced 00:22:31.580 --> 00:22:37.070 align:middle line:84% by a, b, c, d, that means I take these four vertices 00:22:37.070 --> 00:22:38.550 align:middle line:90% and none of the others. 00:22:38.550 --> 00:22:42.950 align:middle line:84% And I take all the edges that are among those four vertices. 00:22:42.950 --> 00:22:44.760 align:middle line:90% So that would be three of them. 00:22:44.760 --> 00:22:48.560 align:middle line:84% These three edges-- a to b, b to c, and b to d-- 00:22:48.560 --> 00:22:53.690 align:middle line:84% are the three edges in my induced subgraph a, b, c, d. 00:22:53.690 --> 00:22:57.110 align:middle line:84% If I just take the subset f and g, 00:22:57.110 --> 00:23:00.260 align:middle line:84% these are all the vertices connected to f, say. 00:23:00.260 --> 00:23:04.040 align:middle line:84% Then my subgraph will just be this picture with these two 00:23:04.040 --> 00:23:07.310 align:middle line:84% vertices and one edge connecting them. 00:23:07.310 --> 00:23:10.770 align:middle line:84% So that's what induced subgraph means. 00:23:10.770 --> 00:23:13.670 align:middle line:84% And so in particular, if I take the induced subgraph 00:23:13.670 --> 00:23:15.763 align:middle line:84% for vertices connected to a particular vertex, 00:23:15.763 --> 00:23:17.430 align:middle line:84% I get what I call a connected component. 00:23:17.430 --> 00:23:20.250 align:middle line:84% So this is one connected component here. 00:23:20.250 --> 00:23:21.680 align:middle line:90% This is another one. 00:23:21.680 --> 00:23:22.937 align:middle line:90% And this is another one. 00:23:22.937 --> 00:23:24.770 align:middle line:84% And those are the three connected components 00:23:24.770 --> 00:23:25.740 align:middle line:90% of the graph. 00:23:25.740 --> 00:23:33.155 align:middle line:84% And in general, the connected components partition the graph. 00:23:33.155 --> 00:23:48.930 align:middle line:90% 00:23:48.930 --> 00:23:54.240 align:middle line:84% Partition means, in this case, that every vertex or edge 00:23:54.240 --> 00:23:57.310 align:middle line:84% is in exactly one connected component. 00:23:57.310 --> 00:23:59.940 align:middle line:90% 00:23:59.940 --> 00:24:03.275 align:middle line:84% So everybody is represented exactly once. 00:24:03.275 --> 00:24:08.670 align:middle line:90% 00:24:08.670 --> 00:24:11.040 align:middle line:84% This is just a fact which we will take as given. 00:24:11.040 --> 00:24:15.360 align:middle line:84% It, in some sense, follows from the symmetry, reflexivity, 00:24:15.360 --> 00:24:18.665 align:middle line:84% and transitivity of the connectivity relation. 00:24:18.665 --> 00:24:21.990 align:middle line:90% 00:24:21.990 --> 00:24:22.830 align:middle line:90% Cool. 00:24:22.830 --> 00:24:24.400 align:middle line:90% So that's connected components. 00:24:24.400 --> 00:24:29.150 align:middle line:84% Again, many possible motivations for connected components. 00:24:29.150 --> 00:24:32.510 align:middle line:84% You would hope that the road network of North America, 00:24:32.510 --> 00:24:35.460 align:middle line:84% the continental-- let's say continental United States is 00:24:35.460 --> 00:24:35.960 align:middle line:90% connected. 00:24:35.960 --> 00:24:39.530 align:middle line:84% It probably isn't because of some weird things, disconnected 00:24:39.530 --> 00:24:42.050 align:middle line:90% things going on with Canada. 00:24:42.050 --> 00:24:47.420 align:middle line:84% Or you might wonder, is the state space of this eight puzzle 00:24:47.420 --> 00:24:48.240 align:middle line:90% connected? 00:24:48.240 --> 00:24:51.125 align:middle line:84% Can you get from any configuration to any other? 00:24:51.125 --> 00:24:52.500 align:middle line:84% We already know the answer is no. 00:24:52.500 --> 00:24:54.500 align:middle line:84% We proved you can't get from an even permutation 00:24:54.500 --> 00:24:56.030 align:middle line:90% to an odd permutation. 00:24:56.030 --> 00:25:01.650 align:middle line:84% So this example happens to be even. 00:25:01.650 --> 00:25:04.380 align:middle line:84% You can see the number of inversions written down here. 00:25:04.380 --> 00:25:06.080 align:middle line:90% It's always even. 00:25:06.080 --> 00:25:11.570 align:middle line:84% And if I take any other permutation that's even, 00:25:11.570 --> 00:25:13.440 align:middle line:84% it turns out it's possible to get there. 00:25:13.440 --> 00:25:14.550 align:middle line:90% We didn't prove that. 00:25:14.550 --> 00:25:17.270 align:middle line:84% What we did prove is that if we take a different configuration 00:25:17.270 --> 00:25:18.900 align:middle line:84% where the number of inversions is odd, 00:25:18.900 --> 00:25:21.380 align:middle line:84% all of these configurations cannot get 00:25:21.380 --> 00:25:24.040 align:middle line:84% to-- they're in a different connected component from this 00:25:24.040 --> 00:25:24.580 align:middle line:90% one. 00:25:24.580 --> 00:25:26.080 align:middle line:84% And so it turns out the eight puzzle 00:25:26.080 --> 00:25:28.220 align:middle line:84% has exactly two connected components, 00:25:28.220 --> 00:25:31.060 align:middle line:84% one for the even permutations, one for the odds. 00:25:31.060 --> 00:25:34.040 align:middle line:84% The Rubik's cube, for example, depending on how you define it, 00:25:34.040 --> 00:25:37.570 align:middle line:90% has 12 connected components. 00:25:37.570 --> 00:25:40.000 align:middle line:84% In general, you might care about how many different types 00:25:40.000 --> 00:25:42.375 align:middle line:84% of states there are for a given puzzle. 00:25:42.375 --> 00:25:45.730 align:middle line:90% 00:25:45.730 --> 00:25:51.520 align:middle line:84% So that's a brief tour of connectivity. 00:25:51.520 --> 00:25:55.420 align:middle line:90% The next concept is-- 00:25:55.420 --> 00:25:57.970 align:middle line:90% maybe I'll start with-- 00:25:57.970 --> 00:26:00.340 align:middle line:84% right now, we're just walking aimlessly 00:26:00.340 --> 00:26:02.030 align:middle line:90% from one vertex to another. 00:26:02.030 --> 00:26:05.200 align:middle line:84% Let's think about closing a loop. 00:26:05.200 --> 00:26:07.820 align:middle line:84% What if I-- I guess it's even more aimless. 00:26:07.820 --> 00:26:08.750 align:middle line:90% I started a vertex. 00:26:08.750 --> 00:26:11.420 align:middle line:84% And what if I want to come back to the same vertex? 00:26:11.420 --> 00:26:13.590 align:middle line:84% This is what we call a closed walk. 00:26:13.590 --> 00:26:28.880 align:middle line:90% 00:26:28.880 --> 00:26:32.010 align:middle line:84% So if a walk happens to start and end at the same vertex, 00:26:32.010 --> 00:26:36.260 align:middle line:84% if v0 equals vk, then we call it closed. 00:26:36.260 --> 00:26:40.070 align:middle line:84% And so in particular, we could try to apply the closed notion 00:26:40.070 --> 00:26:42.080 align:middle line:90% to a path. 00:26:42.080 --> 00:26:43.293 align:middle line:90% That would be kind of nice. 00:26:43.293 --> 00:26:48.770 align:middle line:90% 00:26:48.770 --> 00:26:53.652 align:middle line:84% And we're going to give that a name, which is cycle. 00:26:53.652 --> 00:26:56.510 align:middle line:90% A cycle is roughly-- 00:26:56.510 --> 00:27:00.140 align:middle line:84% this is not going to be quite correct. 00:27:00.140 --> 00:27:04.700 align:middle line:90% Start with a closed path. 00:27:04.700 --> 00:27:07.710 align:middle line:84% Can someone tell me what would be wrong with this definition? 00:27:07.710 --> 00:27:11.630 align:middle line:90% If I say, OK, I want to close-- 00:27:11.630 --> 00:27:13.910 align:middle line:84% so closed is an adjective that means you start and end 00:27:13.910 --> 00:27:15.120 align:middle line:90% at the same vertex. 00:27:15.120 --> 00:27:17.670 align:middle line:84% If I apply it to path, that means a walk, 00:27:17.670 --> 00:27:20.450 align:middle line:84% let's say, from some vertex v to the same vertex 00:27:20.450 --> 00:27:23.190 align:middle line:84% v with no repeated vertices or edges. 00:27:23.190 --> 00:27:24.890 align:middle line:90% Any problems with that? 00:27:24.890 --> 00:27:26.158 align:middle line:90% Yeah. 00:27:26.158 --> 00:27:28.110 align:middle line:84% AUDIENCE: The last part, [INAUDIBLE] 00:27:28.110 --> 00:27:30.337 align:middle line:84% the one you start with, it's going to be repeated. 00:27:30.337 --> 00:27:31.170 align:middle line:90% ERIK DEMAINE: Right. 00:27:31.170 --> 00:27:33.290 align:middle line:84% The one you start with is going to be repeated, 00:27:33.290 --> 00:27:34.620 align:middle line:90% because that's what we wanted. 00:27:34.620 --> 00:27:36.900 align:middle line:84% We want the start and the end to be the same. 00:27:36.900 --> 00:27:38.840 align:middle line:90% So there is no closed path. 00:27:38.840 --> 00:27:41.510 align:middle line:84% Actually, there's kind of one closed path, which 00:27:41.510 --> 00:27:43.730 align:middle line:84% is the path that just starts at a vertex v 00:27:43.730 --> 00:27:44.930 align:middle line:90% and doesn't go anywhere. 00:27:44.930 --> 00:27:46.800 align:middle line:90% That I also want to exclude. 00:27:46.800 --> 00:27:50.520 align:middle line:84% This is not what I want to call a cycle. 00:27:50.520 --> 00:27:51.590 align:middle line:90% So there are two things. 00:27:51.590 --> 00:27:56.190 align:middle line:84% I want a closed path of length more than zero. 00:27:56.190 --> 00:27:58.940 align:middle line:84% You're not allowed to just stay where you are. 00:27:58.940 --> 00:28:05.960 align:middle line:84% And I need to add this exception that the first vertex equals 00:28:05.960 --> 00:28:06.870 align:middle line:90% the last vertex. 00:28:06.870 --> 00:28:09.070 align:middle line:84% I don't want any other vertices to be the same. 00:28:09.070 --> 00:28:12.890 align:middle line:90% 00:28:12.890 --> 00:28:16.680 align:middle line:84% So this is-- now this is sort of a correct definition, 00:28:16.680 --> 00:28:19.290 align:middle line:84% except I'm not a big fan of definitions with exceptions. 00:28:19.290 --> 00:28:22.910 align:middle line:84% So I will write another equivalent and less 00:28:22.910 --> 00:28:25.460 align:middle line:84% controversial definition, which is let's start 00:28:25.460 --> 00:28:28.490 align:middle line:90% with our notion of walk. 00:28:28.490 --> 00:28:31.190 align:middle line:90% And we want a closed one. 00:28:31.190 --> 00:28:33.560 align:middle line:84% And so that implies that the first vertex 00:28:33.560 --> 00:28:34.650 align:middle line:90% equals the last vertex. 00:28:34.650 --> 00:28:39.265 align:middle line:84% And we want no other repeated vertices or edges. 00:28:39.265 --> 00:28:48.710 align:middle line:90% 00:28:48.710 --> 00:28:52.430 align:middle line:84% And whereas with walk, it was-- or with path, it wasn't that big 00:28:52.430 --> 00:28:54.848 align:middle line:84% a deal that I said no repeated edges, 00:28:54.848 --> 00:28:57.390 align:middle line:84% because if you repeat an edge, you also repeat its endpoints. 00:28:57.390 --> 00:29:00.600 align:middle line:84% So repeating an edge implies repeating vertices. 00:29:00.600 --> 00:29:02.940 align:middle line:84% But I put them both in there, just for emphasis. 00:29:02.940 --> 00:29:04.910 align:middle line:90% Here it actually matters-- 00:29:04.910 --> 00:29:06.620 align:middle line:90% say, a path with no-- 00:29:06.620 --> 00:29:09.765 align:middle line:84% a closed walk with no other repeated vertices or edges. 00:29:09.765 --> 00:29:11.640 align:middle line:84% There's another thing that I want to exclude, 00:29:11.640 --> 00:29:14.810 align:middle line:90% which is suppose I have an edge. 00:29:14.810 --> 00:29:16.540 align:middle line:90% And I get my red color. 00:29:16.540 --> 00:29:21.460 align:middle line:90% 00:29:21.460 --> 00:29:23.990 align:middle line:90% And I start at this vertex. 00:29:23.990 --> 00:29:25.160 align:middle line:90% I go to the other one. 00:29:25.160 --> 00:29:26.950 align:middle line:90% And I come back. 00:29:26.950 --> 00:29:30.110 align:middle line:84% You might say, oh, that looks like a length 2 cycle. 00:29:30.110 --> 00:29:31.310 align:middle line:90% But it's not. 00:29:31.310 --> 00:29:33.290 align:middle line:84% I don't want to call this a length 2 cycle. 00:29:33.290 --> 00:29:35.110 align:middle line:84% And it's not allowed by this definition, 00:29:35.110 --> 00:29:39.350 align:middle line:90% because it repeats an edge. 00:29:39.350 --> 00:29:41.530 align:middle line:84% It doesn't repeat any vertices, other than the ones 00:29:41.530 --> 00:29:42.238 align:middle line:90% it's supposed to. 00:29:42.238 --> 00:29:48.190 align:middle line:84% It starts-- the sequence here is u, v, u. 00:29:48.190 --> 00:29:51.250 align:middle line:90% It looks like an emoji. 00:29:51.250 --> 00:29:56.110 align:middle line:84% So there's no repeated vertices, except the first 1 equaling 00:29:56.110 --> 00:29:56.810 align:middle line:90% the last one. 00:29:56.810 --> 00:29:58.100 align:middle line:90% But there is a repeated edge. 00:29:58.100 --> 00:29:59.660 align:middle line:90% And so I don't allow this. 00:29:59.660 --> 00:30:01.900 align:middle line:90% This is not a cycle. 00:30:01.900 --> 00:30:03.910 align:middle line:84% So in fact, in an undirected graph, 00:30:03.910 --> 00:30:07.090 align:middle line:84% a cycle has to have length at least three. 00:30:07.090 --> 00:30:10.780 align:middle line:84% The definition says it has to be at least one. 00:30:10.780 --> 00:30:14.385 align:middle line:84% Oh, I probably need of length greater than 0 here again. 00:30:14.385 --> 00:30:19.190 align:middle line:90% 00:30:19.190 --> 00:30:19.980 align:middle line:90% Cool. 00:30:19.980 --> 00:30:24.990 align:middle line:84% So cycle is sort of the analog of path. 00:30:24.990 --> 00:30:27.613 align:middle line:84% And closed walk is the analog of walk. 00:30:27.613 --> 00:30:29.030 align:middle line:84% Obviously, it's the closed version 00:30:29.030 --> 00:30:30.738 align:middle line:84% where you come back to where you started. 00:30:30.738 --> 00:30:34.850 align:middle line:84% Cycle is the best analog we have to a closed path. 00:30:34.850 --> 00:30:39.090 align:middle line:84% And these are nice in a certain sense, 00:30:39.090 --> 00:30:41.380 align:middle line:90% which I want to talk about. 00:30:41.380 --> 00:30:45.440 align:middle line:90% 00:30:45.440 --> 00:30:54.410 align:middle line:84% So let's talk about a notion which is an Euler walk. 00:30:54.410 --> 00:30:58.963 align:middle line:84% And then we'll get to the notion of a closed walk. 00:30:58.963 --> 00:31:28.390 align:middle line:90% 00:31:28.390 --> 00:31:33.550 align:middle line:84% So an Euler walk is a particular kind of nice walk 00:31:33.550 --> 00:31:37.670 align:middle line:84% that visits all of the edges exactly once. 00:31:37.670 --> 00:31:39.250 align:middle line:90% That's the definition. 00:31:39.250 --> 00:31:42.850 align:middle line:84% Let's think about graphs where you 00:31:42.850 --> 00:31:45.680 align:middle line:84% can take a walk, starting somewhere, ending somewhere, 00:31:45.680 --> 00:31:48.110 align:middle line:84% and visit every edge exactly once. 00:31:48.110 --> 00:31:55.910 align:middle line:84% For example, this graph has such a walk, 00:31:55.910 --> 00:31:57.200 align:middle line:90% the first graph that I drew. 00:31:57.200 --> 00:32:00.730 align:middle line:84% This is why I was fascinated with it, reading some Scientific 00:32:00.730 --> 00:32:04.720 align:middle line:84% American article by Martin Gardner. 00:32:04.720 --> 00:32:08.930 align:middle line:84% You can draw this graph without picking up your pen. 00:32:08.930 --> 00:32:12.140 align:middle line:84% You pen down, you draw, and then you pen up. 00:32:12.140 --> 00:32:13.850 align:middle line:90% And that got all of the edges. 00:32:13.850 --> 00:32:17.192 align:middle line:84% I didn't draw the vertices, but hopefully you can imagine that. 00:32:17.192 --> 00:32:18.900 align:middle line:84% Maybe you should draw the vertices first, 00:32:18.900 --> 00:32:21.840 align:middle line:84% and then you can draw all the edges in one swoop. 00:32:21.840 --> 00:32:25.310 align:middle line:84% So this is kind of a nice notion, 00:32:25.310 --> 00:32:29.720 align:middle line:84% if you like drawing things with pen or pencil or stylus, 00:32:29.720 --> 00:32:30.800 align:middle line:90% or whatever. 00:32:30.800 --> 00:32:32.960 align:middle line:90% It's actually a very old notion. 00:32:32.960 --> 00:32:35.250 align:middle line:90% It goes back to Euler. 00:32:35.250 --> 00:32:36.770 align:middle line:84% Unsurprisingly, this guy Leonhard 00:32:36.770 --> 00:32:41.300 align:middle line:84% Euler, who you know from the number E. That's E for Euler. 00:32:41.300 --> 00:32:45.500 align:middle line:84% In 1736, he drew this rather beautiful diagram 00:32:45.500 --> 00:32:50.210 align:middle line:84% of some islands and bridges in Konigsberg, which then was 00:32:50.210 --> 00:32:52.940 align:middle line:90% in Germany, now is in Russia. 00:32:52.940 --> 00:32:56.090 align:middle line:84% And it has-- this place, the city 00:32:56.090 --> 00:32:59.880 align:middle line:84% is famous for very beautiful bridges, as many cities are. 00:32:59.880 --> 00:33:02.540 align:middle line:84% But this one has a lot of them and some non-trivial, 00:33:02.540 --> 00:33:04.770 align:middle line:90% interesting topology. 00:33:04.770 --> 00:33:06.980 align:middle line:90% And so he wondered-- 00:33:06.980 --> 00:33:11.240 align:middle line:84% the question was posed, can you visit all of the bridges 00:33:11.240 --> 00:33:12.355 align:middle line:90% without any repeats? 00:33:12.355 --> 00:33:14.360 align:middle line:90% It seems natural. 00:33:14.360 --> 00:33:16.100 align:middle line:84% And if you think about it, this is 00:33:16.100 --> 00:33:17.520 align:middle line:90% the vertices are like islands. 00:33:17.520 --> 00:33:19.440 align:middle line:90% The edges are bridges. 00:33:19.440 --> 00:33:23.210 align:middle line:84% And you want to visit all the edges exactly once. 00:33:23.210 --> 00:33:27.230 align:middle line:84% So a natural problem, also, if you're sightseeing. 00:33:27.230 --> 00:33:30.720 align:middle line:84% And just for fun, because I dug up this 1736 paper, 00:33:30.720 --> 00:33:35.180 align:middle line:84% here are some more beautiful figures from that paper, 00:33:35.180 --> 00:33:36.810 align:middle line:84% thinking about all the different cases. 00:33:36.810 --> 00:33:38.570 align:middle line:84% So he characterized-- and this is 00:33:38.570 --> 00:33:41.420 align:middle line:84% like the first paper, or one of the very first papers 00:33:41.420 --> 00:33:42.527 align:middle line:90% in graph theory. 00:33:42.527 --> 00:33:44.360 align:middle line:84% This started the whole world of graph theory 00:33:44.360 --> 00:33:45.380 align:middle line:90% we're talking about. 00:33:45.380 --> 00:33:49.530 align:middle line:84% He's characterizing when this is possible and when it isn't. 00:33:49.530 --> 00:33:53.760 align:middle line:84% It turns out, in this original example, it's not possible. 00:33:53.760 --> 00:33:57.900 align:middle line:84% Today some of these bridges have collapsed or been destroyed, 00:33:57.900 --> 00:33:59.040 align:middle line:90% so maybe it's possible. 00:33:59.040 --> 00:34:05.277 align:middle line:84% I don't know about current day Konigsberg, but that's-- 00:34:05.277 --> 00:34:06.860 align:middle line:84% what we're going to talk about now is, 00:34:06.860 --> 00:34:11.010 align:middle line:90% when do Euler walks exist? 00:34:11.010 --> 00:34:14.020 align:middle line:84% Why was it possible for that house graph, 00:34:14.020 --> 00:34:22.620 align:middle line:90% but not possible for this graph? 00:34:22.620 --> 00:34:25.060 align:middle line:84% This is the complete graph on four vertices. 00:34:25.060 --> 00:34:28.570 align:middle line:84% And you can't draw this in a single path. 00:34:28.570 --> 00:34:32.850 align:middle line:84% Why is my red chalk always the farthest possible point? 00:34:32.850 --> 00:34:34.363 align:middle line:90% So let's try. 00:34:34.363 --> 00:34:36.280 align:middle line:84% This will not be a proof that you can't do it, 00:34:36.280 --> 00:34:40.840 align:middle line:84% but we will shortly prove that you can't do it in one stroke. 00:34:40.840 --> 00:34:46.000 align:middle line:84% So I can go here, here, here, here, here. 00:34:46.000 --> 00:34:48.725 align:middle line:84% It's pretty good, but I missed this edge. 00:34:48.725 --> 00:34:52.560 align:middle line:90% 00:34:52.560 --> 00:34:56.217 align:middle line:90% This edge is uncovered. 00:34:56.217 --> 00:34:58.300 align:middle line:84% And it turns out it's not possible for that graph. 00:34:58.300 --> 00:34:59.950 align:middle line:90% It was possible for this graph. 00:34:59.950 --> 00:35:08.100 align:middle line:84% Even better would be, I guess, in the case-- 00:35:08.100 --> 00:35:11.720 align:middle line:84% in the tourism case, let's go back to Konigsberg for a second. 00:35:11.720 --> 00:35:15.570 align:middle line:84% Suppose your hotel is in island A. You start there. 00:35:15.570 --> 00:35:17.570 align:middle line:84% You'd kind of like to end at the same hotel 00:35:17.570 --> 00:35:20.850 align:middle line:84% without having to repeat a bridge again. 00:35:20.850 --> 00:35:23.575 align:middle line:84% So that would be an Euler closed walk. 00:35:23.575 --> 00:35:31.970 align:middle line:90% 00:35:31.970 --> 00:35:34.910 align:middle line:84% And that we call an Euler tour, just because that's 00:35:34.910 --> 00:35:36.830 align:middle line:90% more common terminology for it. 00:35:36.830 --> 00:35:38.850 align:middle line:84% So that's really where you start. 00:35:38.850 --> 00:35:41.520 align:middle line:84% So closed means that you start and end at the same place. 00:35:41.520 --> 00:35:43.670 align:middle line:84% Euler means you visit every edge exactly once, 00:35:43.670 --> 00:35:45.260 align:middle line:90% every edge of the graph. 00:35:45.260 --> 00:35:47.310 align:middle line:90% So that's even nicer. 00:35:47.310 --> 00:35:51.350 align:middle line:84% It turns out that's not possible in this graph. 00:35:51.350 --> 00:35:56.090 align:middle line:84% If you look at it, I started at this vertex here. 00:35:56.090 --> 00:36:00.800 align:middle line:90% And I ended at this vertex here. 00:36:00.800 --> 00:36:02.882 align:middle line:84% And that was the best I could do. 00:36:02.882 --> 00:36:05.090 align:middle line:84% I couldn't actually start and end at the same vertex. 00:36:05.090 --> 00:36:11.740 align:middle line:84% If, for example, I added another vertex and a couple more edges, 00:36:11.740 --> 00:36:16.120 align:middle line:84% then it would be possible for me to complete the tour like this. 00:36:16.120 --> 00:36:18.960 align:middle line:84% So the new graph I drew here with six vertices 00:36:18.960 --> 00:36:21.550 align:middle line:84% instead of five, that one has an Euler tour. 00:36:21.550 --> 00:36:23.740 align:middle line:84% The original one just had an Euler walk. 00:36:23.740 --> 00:36:26.160 align:middle line:90% And this one has neither. 00:36:26.160 --> 00:36:28.895 align:middle line:84% So how can we characterize which graphs these are? 00:36:28.895 --> 00:36:31.800 align:middle line:90% 00:36:31.800 --> 00:36:34.000 align:middle line:84% So we're not going to read Euler's paper. 00:36:34.000 --> 00:36:37.650 align:middle line:84% It's in Latin-- a bit hard for me to read. 00:36:37.650 --> 00:36:41.285 align:middle line:90% We'll write our own proof. 00:36:41.285 --> 00:36:42.910 align:middle line:84% This will be the main theorem of today. 00:36:42.910 --> 00:36:45.690 align:middle line:84% Though, we have a fun topic afterwards as well. 00:36:45.690 --> 00:36:48.630 align:middle line:84% So first, let's think about the tour case. 00:36:48.630 --> 00:36:50.400 align:middle line:90% That turns out to be cleaner. 00:36:50.400 --> 00:37:01.170 align:middle line:84% So a graph has an Euler tour if and only if it's connected. 00:37:01.170 --> 00:37:03.720 align:middle line:84% That's certainly necessary because we said 00:37:03.720 --> 00:37:05.685 align:middle line:90% it should visit every vertex. 00:37:05.685 --> 00:37:08.060 align:middle line:84% You don't have to include this as part of the definition, 00:37:08.060 --> 00:37:11.920 align:middle line:84% but it makes the characterization cleaner. 00:37:11.920 --> 00:37:20.445 align:middle line:84% And we also need that every vertex has even degree. 00:37:20.445 --> 00:37:24.550 align:middle line:90% 00:37:24.550 --> 00:37:26.960 align:middle line:84% For an Euler tour, for a closed walk, 00:37:26.960 --> 00:37:29.150 align:middle line:84% we need every vertex to have even degree. 00:37:29.150 --> 00:37:31.220 align:middle line:84% Let's check that in our examples. 00:37:31.220 --> 00:37:33.550 align:middle line:84% This graph, remember, the degree of a vertex 00:37:33.550 --> 00:37:35.300 align:middle line:84% is how many edges are incident to it. 00:37:35.300 --> 00:37:40.390 align:middle line:84% So the degree of this vertex is 3. 00:37:40.390 --> 00:37:42.020 align:middle line:90% The degree of this vertex is 3. 00:37:42.020 --> 00:37:45.040 align:middle line:84% This graph is symmetric, so all the vertices have degree 3. 00:37:45.040 --> 00:37:49.630 align:middle line:84% 3 is an odd number, and not even. 00:37:49.630 --> 00:37:51.770 align:middle line:90% In this graph, this is degree 2. 00:37:51.770 --> 00:37:52.910 align:middle line:90% This is degree 4. 00:37:52.910 --> 00:37:53.810 align:middle line:90% This is degree 4. 00:37:53.810 --> 00:37:55.300 align:middle line:90% So far, everything is even. 00:37:55.300 --> 00:37:57.260 align:middle line:90% This one is degree 3. 00:37:57.260 --> 00:37:59.200 align:middle line:84% This one is degree 3 in the original graph. 00:37:59.200 --> 00:38:01.900 align:middle line:84% If I add this dotted part, then I 00:38:01.900 --> 00:38:06.180 align:middle line:84% convert this into degree 4, and degree 4, and degree 2 00:38:06.180 --> 00:38:07.000 align:middle line:90% down here. 00:38:07.000 --> 00:38:08.920 align:middle line:84% And so then all the vertices have even degree. 00:38:08.920 --> 00:38:12.930 align:middle line:84% So it seems to be working, but let's not do a proof by example. 00:38:12.930 --> 00:38:14.675 align:middle line:90% Let's prove it in general. 00:38:14.675 --> 00:38:23.400 align:middle line:90% 00:38:23.400 --> 00:38:27.340 align:middle line:84% Again, the top level statement here is an if and only if. 00:38:27.340 --> 00:38:29.770 align:middle line:84% So we need to prove each direction separately. 00:38:29.770 --> 00:38:32.140 align:middle line:84% Let's start with the easy direction, 00:38:32.140 --> 00:38:33.900 align:middle line:90% which is left to right. 00:38:33.900 --> 00:38:37.410 align:middle line:90% So if I have an Euler tour-- 00:38:37.410 --> 00:38:39.420 align:middle line:90% let's give it a name. 00:38:39.420 --> 00:38:51.840 align:middle line:84% Consider an Euler tour v0 to v1 to vk. 00:38:51.840 --> 00:38:57.155 align:middle line:84% And vk equals v0 because it's a closed walk. 00:38:57.155 --> 00:38:59.830 align:middle line:90% 00:38:59.830 --> 00:39:01.803 align:middle line:90% Now I want to look at-- 00:39:01.803 --> 00:39:04.450 align:middle line:90% 00:39:04.450 --> 00:39:17.700 align:middle line:84% let's basically count edges incident to vi or some vertex. 00:39:17.700 --> 00:39:21.700 align:middle line:90% 00:39:21.700 --> 00:39:23.690 align:middle line:84% I think that's an unnecessary sentence. 00:39:23.690 --> 00:39:27.580 align:middle line:90% Let me rewind. 00:39:27.580 --> 00:39:33.520 align:middle line:90% Let's look at each visit. 00:39:33.520 --> 00:39:45.110 align:middle line:84% vi minus 1 to vi to vi plus 1 to a vertex vi. 00:39:45.110 --> 00:39:48.130 align:middle line:84% So in this path, I'm just looking 00:39:48.130 --> 00:39:52.090 align:middle line:84% at a little piece of the path of two consecutive edges from vi 00:39:52.090 --> 00:39:54.430 align:middle line:90% minus 1 to vi to vi plus 1. 00:39:54.430 --> 00:39:58.840 align:middle line:84% And I'd like this to work even with wraparound. 00:39:58.840 --> 00:40:04.120 align:middle line:84% Because this is a closed tour, we start at v0 and we end at v0. 00:40:04.120 --> 00:40:06.990 align:middle line:90% So we do some stuff. 00:40:06.990 --> 00:40:09.900 align:middle line:90% And then we end up back at v0. 00:40:09.900 --> 00:40:13.980 align:middle line:84% And so I'd like to just consider three consecutive vertices 00:40:13.980 --> 00:40:15.400 align:middle line:90% on this tour. 00:40:15.400 --> 00:40:15.900 align:middle line:90% Sorry. 00:40:15.900 --> 00:40:17.460 align:middle line:90% Wrong directions. 00:40:17.460 --> 00:40:20.890 align:middle line:90% This way, including possibly v0. 00:40:20.890 --> 00:40:29.470 align:middle line:84% So to do that, I want to treat this modulo, k plus 1, I guess. 00:40:29.470 --> 00:40:31.860 align:middle line:84% And this also modulo or no modulo 00:40:31.860 --> 00:40:37.000 align:middle line:84% k, actually, because vk equals v0. 00:40:37.000 --> 00:40:40.785 align:middle line:84% So if I work modulo k with these indices, for every vertex i, 00:40:40.785 --> 00:40:43.140 align:middle line:84% I look at the one previous possibly wrapping around 00:40:43.140 --> 00:40:47.520 align:middle line:84% to vk minus 1 or vi plus 1, possibly wrapping around 00:40:47.520 --> 00:40:52.140 align:middle line:84% back to v0, then this is a well-defined notion 00:40:52.140 --> 00:40:55.150 align:middle line:90% for every vertex on this visit. 00:40:55.150 --> 00:40:58.650 align:middle line:84% When I look at this, that visit to vi, 00:40:58.650 --> 00:41:01.090 align:middle line:84% in particular, that's visiting a vertex. 00:41:01.090 --> 00:41:13.400 align:middle line:84% And it also includes or visits two edges incident to vi, 00:41:13.400 --> 00:41:15.152 align:middle line:84% sort of "what goes in must come out." 00:41:15.152 --> 00:41:16.610 align:middle line:84% So if I'm looking at a vertex here, 00:41:16.610 --> 00:41:21.710 align:middle line:90% vi, this is the previous one. 00:41:21.710 --> 00:41:23.990 align:middle line:90% Oh, this is the previous one. 00:41:23.990 --> 00:41:27.030 align:middle line:90% i minus 1 to vi plus 1. 00:41:27.030 --> 00:41:30.540 align:middle line:84% But by visiting vi, I visited one edge coming into vi. 00:41:30.540 --> 00:41:33.210 align:middle line:84% And I visited one edge coming out of vi. 00:41:33.210 --> 00:41:35.090 align:middle line:90% And so that's two. 00:41:35.090 --> 00:41:38.600 align:middle line:84% And maybe this path visits this vertex vi, again, 00:41:38.600 --> 00:41:40.140 align:middle line:90% as some other vertex vj. 00:41:40.140 --> 00:41:42.000 align:middle line:84% But every time it visits a vertex, 00:41:42.000 --> 00:41:44.900 align:middle line:84% it visits two edges incident to it. 00:41:44.900 --> 00:41:48.590 align:middle line:84% So that means the total number of edges visited 00:41:48.590 --> 00:41:51.560 align:middle line:84% is going to be even, because I just keep adding two, and then 00:41:51.560 --> 00:41:53.010 align:middle line:90% two more and two more. 00:41:53.010 --> 00:41:57.240 align:middle line:84% So total number of edges, I guess, 00:41:57.240 --> 00:42:10.760 align:middle line:84% visited edges incident to any vertex must be even. 00:42:10.760 --> 00:42:14.520 align:middle line:90% 00:42:14.520 --> 00:42:18.840 align:middle line:84% And how many edges incident to that vertex do I visit? 00:42:18.840 --> 00:42:21.840 align:middle line:84% Well, all of them, because an Euler walk 00:42:21.840 --> 00:42:25.800 align:middle line:84% has to visit every edge exactly once by the end. 00:42:25.800 --> 00:42:28.255 align:middle line:84% So this total number of visited edges incident 00:42:28.255 --> 00:42:30.463 align:middle line:84% to a vertex, that's exactly the degree of the vertex. 00:42:30.463 --> 00:42:33.720 align:middle line:90% 00:42:33.720 --> 00:42:36.250 align:middle line:84% It's exactly the total number of incident edges. 00:42:36.250 --> 00:42:38.890 align:middle line:84% So this shows that if you have an Euler tour, 00:42:38.890 --> 00:42:42.630 align:middle line:90% all the degrees must be even. 00:42:42.630 --> 00:42:44.080 align:middle line:90% What goes in must come out. 00:42:44.080 --> 00:42:46.560 align:middle line:84% It also has to be connected, just because it 00:42:46.560 --> 00:42:48.070 align:middle line:90% has to visit every vertex. 00:42:48.070 --> 00:42:50.890 align:middle line:84% And so by visiting every vertex in particular, 00:42:50.890 --> 00:42:53.410 align:middle line:84% that is a walk between any pair of vertices. 00:42:53.410 --> 00:42:56.580 align:middle line:84% Or you take pieces of that walk, and you can go from any vertex 00:42:56.580 --> 00:42:58.020 align:middle line:90% to any other. 00:42:58.020 --> 00:43:00.060 align:middle line:90% So that's one direction. 00:43:00.060 --> 00:43:01.230 align:middle line:90% That's the easy one. 00:43:01.230 --> 00:43:03.640 align:middle line:90% Let's do the hard one. 00:43:03.640 --> 00:43:36.830 align:middle line:90% 00:43:36.830 --> 00:43:38.580 align:middle line:84% And although I won't be explicit about it, 00:43:38.580 --> 00:43:41.510 align:middle line:84% this proof also, in some sense, gives you a strategy 00:43:41.510 --> 00:43:43.190 align:middle line:90% for finding these tours. 00:43:43.190 --> 00:43:44.570 align:middle line:90% They're not too hard to find. 00:43:44.570 --> 00:43:47.430 align:middle line:90% 00:43:47.430 --> 00:43:47.973 align:middle line:90% Next page. 00:43:47.973 --> 00:43:50.720 align:middle line:90% 00:43:50.720 --> 00:43:55.100 align:middle line:84% So now we want to go from right to left. 00:43:55.100 --> 00:44:01.230 align:middle line:84% So we get to assume that the graph is connected 00:44:01.230 --> 00:44:03.980 align:middle line:84% and that all the vertices have even degree. 00:44:03.980 --> 00:44:07.830 align:middle line:90% 00:44:07.830 --> 00:44:11.790 align:middle line:90% So we say, all degrees even. 00:44:11.790 --> 00:44:17.320 align:middle line:84% And we want to show that the graph has an Euler tour. 00:44:17.320 --> 00:44:20.670 align:middle line:84% For this proof-- so there's one clever idea, 00:44:20.670 --> 00:44:22.356 align:middle line:84% and then the rest is just mechanical. 00:44:22.356 --> 00:44:25.320 align:middle line:90% 00:44:25.320 --> 00:44:28.410 align:middle line:84% Whereas before in this proof that I just erased, in order 00:44:28.410 --> 00:44:30.570 align:middle line:84% to convert a walk into a path, we 00:44:30.570 --> 00:44:33.780 align:middle line:84% took the shortest path, the shortest walk, and the shortest 00:44:33.780 --> 00:44:36.060 align:middle line:84% walk ended up having to be a path. 00:44:36.060 --> 00:44:38.860 align:middle line:84% For an Euler tour, we kind of want to do the opposite. 00:44:38.860 --> 00:44:43.650 align:middle line:84% I want the longest walk that doesn't repeat any edges. 00:44:43.650 --> 00:44:45.150 align:middle line:84% The longest walk that doesn't repeat 00:44:45.150 --> 00:44:47.880 align:middle line:84% any edges, that turns out it will visit every edge 00:44:47.880 --> 00:44:50.100 align:middle line:90% exactly once. 00:44:50.100 --> 00:44:52.260 align:middle line:84% So for that, it's maybe useful to define 00:44:52.260 --> 00:44:54.735 align:middle line:90% another term, which is trail. 00:44:54.735 --> 00:44:56.360 align:middle line:84% We're really not going to use this much 00:44:56.360 --> 00:44:58.500 align:middle line:84% in the class, so feel free to forget it, 00:44:58.500 --> 00:45:03.290 align:middle line:84% other than in the context of this proof, which is 00:45:03.290 --> 00:45:06.290 align:middle line:90% a walk with no repeated edges. 00:45:06.290 --> 00:45:10.440 align:middle line:84% So in other words, it visits every edge at most once. 00:45:10.440 --> 00:45:13.800 align:middle line:84% And then we're going to take the longest trail. 00:45:13.800 --> 00:45:16.770 align:middle line:84% So subject to every edge being visited at most once, 00:45:16.770 --> 00:45:19.880 align:middle line:84% I want to visit as many edges as possible. 00:45:19.880 --> 00:45:25.410 align:middle line:84% And that turns out it will be an Euler walk. 00:45:25.410 --> 00:45:35.240 align:middle line:84% So let's take the longest trail in the graph. 00:45:35.240 --> 00:45:38.930 align:middle line:84% And I'll call it T. And it's going 00:45:38.930 --> 00:45:44.290 align:middle line:84% to, say, start at vertex v0, go to v1, and go to vk. 00:45:44.290 --> 00:45:48.650 align:middle line:90% 00:45:48.650 --> 00:45:50.960 align:middle line:84% I need to prove two things about this trail. 00:45:50.960 --> 00:45:56.740 align:middle line:84% In order to be an Euler tour, it has to be Euler. 00:45:56.740 --> 00:45:58.530 align:middle line:84% It has to visit every edge exactly once, 00:45:58.530 --> 00:46:00.770 align:middle line:90% and it has to be closed. 00:46:00.770 --> 00:46:03.640 align:middle line:84% It turns out it's helpful to prove closed first. 00:46:03.640 --> 00:46:11.330 align:middle line:84% So first claim is that this trail T is closed. 00:46:11.330 --> 00:46:18.450 align:middle line:90% 00:46:18.450 --> 00:46:20.770 align:middle line:90% So we start at v0. 00:46:20.770 --> 00:46:21.940 align:middle line:90% We end at vk. 00:46:21.940 --> 00:46:24.840 align:middle line:90% Closed means that v0 equals vk. 00:46:24.840 --> 00:46:28.270 align:middle line:84% So if we do-- let's do a proof by contradiction. 00:46:28.270 --> 00:46:37.530 align:middle line:90% 00:46:37.530 --> 00:46:43.400 align:middle line:90% Suppose v0 does not equal vk. 00:46:43.400 --> 00:46:46.200 align:middle line:90% 00:46:46.200 --> 00:46:48.690 align:middle line:84% And I want to get a contradiction, which basically 00:46:48.690 --> 00:46:51.830 align:middle line:84% means that somehow I want to show the trail was not actually 00:46:51.830 --> 00:46:55.460 align:middle line:90% longest in that case. 00:46:55.460 --> 00:46:57.480 align:middle line:90% So again, we have some-- 00:46:57.480 --> 00:46:59.600 align:middle line:90% we start at v0. 00:46:59.600 --> 00:47:01.610 align:middle line:90% We do some wiggly path. 00:47:01.610 --> 00:47:06.140 align:middle line:84% And we end at vk, and supposing these are different. 00:47:06.140 --> 00:47:08.670 align:middle line:84% I'd like to claim that I can make this walk longer. 00:47:08.670 --> 00:47:12.830 align:middle line:84% In particular, I claim that there's an edge out of vk 00:47:12.830 --> 00:47:15.560 align:middle line:84% that I haven't visited yet, which means 00:47:15.560 --> 00:47:19.170 align:middle line:90% I could extend the path by one. 00:47:19.170 --> 00:47:20.520 align:middle line:90% That's my claim. 00:47:20.520 --> 00:47:24.180 align:middle line:84% And I'm going to do that by counting the edges at vk. 00:47:24.180 --> 00:47:26.450 align:middle line:90% How many have I used? 00:47:26.450 --> 00:47:29.270 align:middle line:84% Very similar to this proof actually-- 00:47:29.270 --> 00:47:33.300 align:middle line:84% so vk, so this is another sequence from v0 to vk, 00:47:33.300 --> 00:47:35.940 align:middle line:84% except now we have not equals at the end. 00:47:35.940 --> 00:47:42.120 align:middle line:84% So let's think about all the visits to vk by this-- 00:47:42.120 --> 00:47:43.620 align:middle line:84% I should really be pointing at this. 00:47:43.620 --> 00:47:44.880 align:middle line:90% So here's vk. 00:47:44.880 --> 00:47:46.830 align:middle line:84% We know there's a visit at the end. 00:47:46.830 --> 00:47:48.390 align:middle line:84% Possibly it's visited in between. 00:47:48.390 --> 00:47:50.550 align:middle line:90% Maybe some of the vi's equal vk. 00:47:50.550 --> 00:47:53.830 align:middle line:84% We know v0 doesn't equal vk, but some of the others might. 00:47:53.830 --> 00:47:58.210 align:middle line:84% Every time I visit one of those, it looks exactly like this. 00:47:58.210 --> 00:48:04.170 align:middle line:84% In other words, I visit two edges incident to vk. 00:48:04.170 --> 00:48:17.550 align:middle line:84% So let's say for every vi that's equal to vk we visit two edges 00:48:17.550 --> 00:48:22.800 align:middle line:90% incident to vk. 00:48:22.800 --> 00:48:26.850 align:middle line:84% So that's an even number in total. 00:48:26.850 --> 00:48:31.530 align:middle line:90% This is for i not equal to k. 00:48:31.530 --> 00:48:34.390 align:middle line:84% But then there's this last visit at the end. 00:48:34.390 --> 00:48:36.940 align:middle line:84% And this one is special because we stopped. 00:48:36.940 --> 00:48:39.970 align:middle line:84% There's no edge after vk because this doesn't loop around. 00:48:39.970 --> 00:48:43.980 align:middle line:84% So there's an edge before coming into vk, but none after. 00:48:43.980 --> 00:48:54.210 align:middle line:84% So that means, I guess, for i equal k, for that visit to vk, 00:48:54.210 --> 00:49:00.730 align:middle line:84% we visit one edge incident to vk. 00:49:00.730 --> 00:49:03.740 align:middle line:90% 00:49:03.740 --> 00:49:08.390 align:middle line:84% So in this trail how many edges incident to vk did we visit? 00:49:08.390 --> 00:49:11.810 align:middle line:84% An odd number, because we had an even number for all 00:49:11.810 --> 00:49:15.680 align:middle line:84% the earlier visits, and then we had one more at the end, so even 00:49:15.680 --> 00:49:17.720 align:middle line:90% plus one is odd. 00:49:17.720 --> 00:49:31.220 align:middle line:84% So that means T visits an odd number of edges incident to vk. 00:49:31.220 --> 00:49:33.920 align:middle line:84% But we assume that the degrees are even. 00:49:33.920 --> 00:49:36.080 align:middle line:84% And that means there's at least one more 00:49:36.080 --> 00:49:40.880 align:middle line:90% edge that we assumed here. 00:49:40.880 --> 00:49:42.120 align:middle line:90% All degrees are even. 00:49:42.120 --> 00:49:45.000 align:middle line:84% So in particular, vk's degree is even. 00:49:45.000 --> 00:49:53.080 align:middle line:84% And so that means there exists an edge incident to vk 00:49:53.080 --> 00:49:59.230 align:middle line:90% not visited by T. 00:49:59.230 --> 00:50:14.710 align:middle line:84% But then we can extend T. It can go from v0 up to vk, and then 00:50:14.710 --> 00:50:18.460 align:middle line:90% plus whatever one more edge. 00:50:18.460 --> 00:50:23.080 align:middle line:84% So if I call this edge here, vk comma vk 00:50:23.080 --> 00:50:26.510 align:middle line:84% plus 1, because I haven't defined a vk plus 1 yet, 00:50:26.510 --> 00:50:32.920 align:middle line:84% then this will become a strictly longer trail 00:50:32.920 --> 00:50:37.450 align:middle line:84% because that edge wasn't visited before, but now it is. 00:50:37.450 --> 00:50:40.390 align:middle line:90% This is a longer trail-- 00:50:40.390 --> 00:50:41.590 align:middle line:90% contradiction. 00:50:41.590 --> 00:50:46.360 align:middle line:90% 00:50:46.360 --> 00:50:47.400 align:middle line:90% What did it contradict? 00:50:47.400 --> 00:50:51.690 align:middle line:84% I assumed here that v0 does not equal vk. 00:50:51.690 --> 00:50:53.850 align:middle line:84% So in fact, we learned that v0 equals vk. 00:50:53.850 --> 00:50:55.020 align:middle line:90% In other words, T is closed. 00:50:55.020 --> 00:50:56.520 align:middle line:90% So I proved this claim. 00:50:56.520 --> 00:50:58.920 align:middle line:84% I haven't yet proved the whole thing I want. 00:50:58.920 --> 00:51:05.390 align:middle line:84% So at this point, I've proved that my walk is closed. 00:51:05.390 --> 00:51:08.483 align:middle line:84% But I still need to prove that it's Euler. 00:51:08.483 --> 00:51:10.150 align:middle line:84% Those are the two things I need to show. 00:51:10.150 --> 00:51:13.940 align:middle line:90% 00:51:13.940 --> 00:51:15.685 align:middle line:90% So let's do that next. 00:51:15.685 --> 00:51:36.740 align:middle line:90% 00:51:36.740 --> 00:51:41.455 align:middle line:84% So second claim-- and this will be the last one for this proof. 00:51:41.455 --> 00:51:47.350 align:middle line:90% 00:51:47.350 --> 00:51:50.620 align:middle line:90% So T visits every edge. 00:51:50.620 --> 00:51:57.670 align:middle line:84% I'm going to give it a name, u, v, in the edge set of the graph. 00:51:57.670 --> 00:52:02.360 align:middle line:84% So this is what I want, because, by definition, 00:52:02.360 --> 00:52:05.600 align:middle line:84% because I started with a trail, it has no repeated edges. 00:52:05.600 --> 00:52:07.222 align:middle line:84% So if this is every edge at most once, 00:52:07.222 --> 00:52:09.430 align:middle line:84% and this is saying that it visits every edge at least 00:52:09.430 --> 00:52:12.880 align:middle line:84% once, and therefore, it, in fact, 00:52:12.880 --> 00:52:14.510 align:middle line:90% visits every edge exactly once. 00:52:14.510 --> 00:52:17.230 align:middle line:84% And that's the definition of Euler. 00:52:17.230 --> 00:52:19.215 align:middle line:90% So that's what I need to prove. 00:52:19.215 --> 00:52:23.590 align:middle line:90% 00:52:23.590 --> 00:52:27.430 align:middle line:90% Again, proof by contradiction-- 00:52:27.430 --> 00:52:31.586 align:middle line:84% not sure why, but today has lots of contradictions in it. 00:52:31.586 --> 00:52:34.750 align:middle line:90% It seems to be the easiest way. 00:52:34.750 --> 00:52:40.520 align:middle line:84% Suppose not-- this is a very concise way to say, 00:52:40.520 --> 00:52:44.210 align:middle line:84% suppose that it doesn't visit some edge u, v, and e. 00:52:44.210 --> 00:52:46.695 align:middle line:90% I'll write it out. 00:52:46.695 --> 00:53:00.570 align:middle line:84% It doesn't visit some edge u, v, and e. 00:53:00.570 --> 00:53:02.300 align:middle line:84% Again, I want to get a contradiction 00:53:02.300 --> 00:53:04.950 align:middle line:90% to T being the longest trail. 00:53:04.950 --> 00:53:08.840 align:middle line:84% So I'm going to somehow try to make T longer. 00:53:08.840 --> 00:53:14.250 align:middle line:84% So it doesn't visit u, v, but it visits some stuff, v0 to vk. 00:53:14.250 --> 00:53:17.390 align:middle line:84% So let's try to connect them together. 00:53:17.390 --> 00:53:24.840 align:middle line:84% So we've got-- here's the edge u, v that's not covered. 00:53:24.840 --> 00:53:28.820 align:middle line:90% And then over here, we have v0. 00:53:28.820 --> 00:53:32.820 align:middle line:84% Some crazy path that can revisit vertices and whatever. 00:53:32.820 --> 00:53:33.390 align:middle line:90% It's a walk. 00:53:33.390 --> 00:53:33.890 align:middle line:90% Sorry. 00:53:33.890 --> 00:53:36.110 align:middle line:84% I keep saying path when I mean walk. 00:53:36.110 --> 00:53:38.270 align:middle line:84% vk-- oh, actually, at this point, 00:53:38.270 --> 00:53:40.380 align:middle line:90% we know that vk equals v0. 00:53:40.380 --> 00:53:45.370 align:middle line:84% So it ends where it started, but somehow it 00:53:45.370 --> 00:53:46.400 align:middle line:90% doesn't use this edge. 00:53:46.400 --> 00:53:47.690 align:middle line:90% It might use these vertices. 00:53:47.690 --> 00:53:49.270 align:middle line:90% It might not. 00:53:49.270 --> 00:53:58.120 align:middle line:84% But what I'd like to do is connect v 00:53:58.120 --> 00:54:02.470 align:middle line:90% to v0, which is also vk. 00:54:02.470 --> 00:54:03.615 align:middle line:90% Why can I do that? 00:54:03.615 --> 00:54:06.700 align:middle line:90% 00:54:06.700 --> 00:54:08.260 align:middle line:90% How do I know? 00:54:08.260 --> 00:54:09.700 align:middle line:90% I claim there's a walk. 00:54:09.700 --> 00:54:15.070 align:middle line:90% 00:54:15.070 --> 00:54:15.710 align:middle line:90% Yeah. 00:54:15.710 --> 00:54:17.418 align:middle line:84% AUDIENCE: Because the graph is connected. 00:54:17.418 --> 00:54:20.680 align:middle line:84% ERIK DEMAINE: Because we assume the graph is connected. 00:54:20.680 --> 00:54:22.700 align:middle line:90% Way up here-- assume connected. 00:54:22.700 --> 00:54:24.755 align:middle line:84% That's part of this characterization. 00:54:24.755 --> 00:54:26.380 align:middle line:84% So when we're going from right to left, 00:54:26.380 --> 00:54:28.047 align:middle line:84% we get to assume the graph is connected. 00:54:28.047 --> 00:54:30.230 align:middle line:84% Therefore, all pairs of vertices have a walk. 00:54:30.230 --> 00:54:32.360 align:middle line:84% So in particular, these two vertices have a walk. 00:54:32.360 --> 00:54:34.060 align:middle line:90% Now, it's not part of the trail. 00:54:34.060 --> 00:54:35.090 align:middle line:90% It's just another walk. 00:54:35.090 --> 00:54:37.570 align:middle line:84% So this thing over here is T. And this thing 00:54:37.570 --> 00:54:40.330 align:middle line:90% I'm going to call w. 00:54:40.330 --> 00:54:42.520 align:middle line:90% So let's write down some words. 00:54:42.520 --> 00:54:49.252 align:middle line:90% 00:54:49.252 --> 00:54:52.490 align:middle line:90% g is connected. 00:54:52.490 --> 00:54:59.180 align:middle line:90% So there exists a walk w-- 00:54:59.180 --> 00:55:00.743 align:middle line:90% or sorry. 00:55:00.743 --> 00:55:01.910 align:middle line:90% You'll notice I called it x. 00:55:01.910 --> 00:55:07.942 align:middle line:84% I guess x for connection from v to vk. 00:55:07.942 --> 00:55:18.440 align:middle line:90% 00:55:18.440 --> 00:55:22.730 align:middle line:84% I think I really want this to be a path, if you don't mind. 00:55:22.730 --> 00:55:25.770 align:middle line:84% We know for connectivity's sake this is the same thing. 00:55:25.770 --> 00:55:27.800 align:middle line:84% If there's a walk, there's also a path. 00:55:27.800 --> 00:55:30.110 align:middle line:84% Now, let's assume, furthermore, this 00:55:30.110 --> 00:55:34.710 align:middle line:84% is a path, in particular because I want it to be a trail. 00:55:34.710 --> 00:55:38.600 align:middle line:84% So I don't want it to repeat any edges. 00:55:38.600 --> 00:55:41.150 align:middle line:84% And a path won't repeat vertices or edges. 00:55:41.150 --> 00:55:42.730 align:middle line:90% So that's a little better. 00:55:42.730 --> 00:55:45.110 align:middle line:90% It's even better. 00:55:45.110 --> 00:55:48.230 align:middle line:84% And I'm going to give this path some words. 00:55:48.230 --> 00:55:51.200 align:middle line:84% So it starts at v, which we're going to call x0. 00:55:51.200 --> 00:56:01.173 align:middle line:84% And then it goes to x1, and so on to xl, I guess, which is vk. 00:56:01.173 --> 00:56:04.340 align:middle line:90% 00:56:04.340 --> 00:56:08.200 align:middle line:90% It ends at vk, which is also v0. 00:56:08.200 --> 00:56:13.930 align:middle line:84% Now, I claim I can make the path longer. 00:56:13.930 --> 00:56:14.540 align:middle line:90% Sorry. 00:56:14.540 --> 00:56:18.530 align:middle line:84% I can make the original trail T longer as follows. 00:56:18.530 --> 00:56:20.150 align:middle line:90% I'm going to start at u. 00:56:20.150 --> 00:56:24.130 align:middle line:84% I'm going to go to v. I'm going to follow x. 00:56:24.130 --> 00:56:27.260 align:middle line:90% And now I'm at the loop. 00:56:27.260 --> 00:56:28.960 align:middle line:90% And then I want to loop around. 00:56:28.960 --> 00:56:32.500 align:middle line:84% That doesn't quite work, because, plausibly, 00:56:32.500 --> 00:56:36.730 align:middle line:84% some of the vertices along x here, maybe 00:56:36.730 --> 00:56:40.720 align:middle line:84% there's some vertex here, which is actually on this path-- 00:56:40.720 --> 00:56:43.950 align:middle line:90% sorry-- on the trail T. 00:56:43.950 --> 00:56:45.520 align:middle line:84% I want to give this vertex a name. 00:56:45.520 --> 00:56:47.200 align:middle line:90% So the whole path is called x. 00:56:47.200 --> 00:56:51.990 align:middle line:84% This vertex I want to call x sub i. 00:56:51.990 --> 00:56:57.960 align:middle line:84% So let's let x sub i, which is also 00:56:57.960 --> 00:57:08.610 align:middle line:84% some v sub j be the first vertex of x that's 00:57:08.610 --> 00:57:16.330 align:middle line:90% on T. We start at some vertex. 00:57:16.330 --> 00:57:19.390 align:middle line:84% It might-- it may be the very first vertex v is actually on T. 00:57:19.390 --> 00:57:23.130 align:middle line:84% So then x sub i is actually x sub 0. 00:57:23.130 --> 00:57:25.050 align:middle line:84% But we know eventually we get on to T, 00:57:25.050 --> 00:57:27.840 align:middle line:84% because the last vertex on this path x-- 00:57:27.840 --> 00:57:29.440 align:middle line:90% this is the path x. 00:57:29.440 --> 00:57:34.920 align:middle line:84% We know the last vertex is vk, and that's on T. 00:57:34.920 --> 00:57:37.350 align:middle line:84% So somewhere along the way-- it could be the first vertex; 00:57:37.350 --> 00:57:39.808 align:middle line:84% it could be the last one; it could be anywhere in between-- 00:57:39.808 --> 00:57:43.620 align:middle line:84% we intersect T. And that's going to be my path. 00:57:43.620 --> 00:57:49.470 align:middle line:84% So I'm going to start at u, go to v, follow x until I reach T, 00:57:49.470 --> 00:57:52.040 align:middle line:90% and now T is a loop-- 00:57:52.040 --> 00:57:53.880 align:middle line:90% is a closed walk. 00:57:53.880 --> 00:58:01.010 align:middle line:84% And I'm going to follow that walk as it was before. 00:58:01.010 --> 00:58:02.793 align:middle line:84% I wish I didn't draw this quite so long-- 00:58:02.793 --> 00:58:05.870 align:middle line:90% 00:58:05.870 --> 00:58:07.295 align:middle line:90% exactly what I had before. 00:58:07.295 --> 00:58:10.380 align:middle line:84% It visits exactly the same edges I had before. 00:58:10.380 --> 00:58:12.390 align:middle line:90% We walk, so we can stop here. 00:58:12.390 --> 00:58:18.620 align:middle line:84% I claim that this red walk that I drew is a longer trail. 00:58:18.620 --> 00:58:19.680 align:middle line:90% Why is it longer? 00:58:19.680 --> 00:58:23.520 align:middle line:84% Because it has this edge now, u, v, which it didn't before. 00:58:23.520 --> 00:58:25.500 align:middle line:84% And it has all the edges we had before. 00:58:25.500 --> 00:58:27.907 align:middle line:84% Maybe it has some more here, but it also might not. 00:58:27.907 --> 00:58:29.490 align:middle line:84% Maybe these two vertices are the same, 00:58:29.490 --> 00:58:31.860 align:middle line:84% but we've got at least plus 1 here. 00:58:31.860 --> 00:58:34.470 align:middle line:84% And we visited all the edges we had before. 00:58:34.470 --> 00:58:38.130 align:middle line:84% And because of the way we chose xi, all of these edges 00:58:38.130 --> 00:58:39.960 align:middle line:90% were not used before. 00:58:39.960 --> 00:58:43.860 align:middle line:90% So it's still a trail. 00:58:43.860 --> 00:58:46.710 align:middle line:90% And so I guess what do I write? 00:58:46.710 --> 00:58:52.740 align:middle line:84% u to v, which is also known as x0. 00:58:52.740 --> 00:58:55.308 align:middle line:90% Then we go to xi. 00:58:55.308 --> 00:58:58.770 align:middle line:90% xi equals vj. 00:58:58.770 --> 00:59:05.520 align:middle line:84% And then we go to vj plus 1, and so on, around T. In the notes, 00:59:05.520 --> 00:59:06.640 align:middle line:90% I wrote it explicitly. 00:59:06.640 --> 00:59:09.480 align:middle line:90% You go from vj up to vk. 00:59:09.480 --> 00:59:10.560 align:middle line:90% And then you loop around. 00:59:10.560 --> 00:59:14.530 align:middle line:84% And you go from v0 up to vj again. 00:59:14.530 --> 00:59:20.190 align:middle line:84% And that will be all the edges we had before. 00:59:20.190 --> 00:59:21.883 align:middle line:90% So it's longer. 00:59:21.883 --> 00:59:25.080 align:middle line:90% 00:59:25.080 --> 00:59:28.320 align:middle line:84% And this will be a trail because we didn't-- 00:59:28.320 --> 00:59:32.460 align:middle line:84% all the edges that we added were not in the original trail T. 00:59:32.460 --> 00:59:35.080 align:middle line:84% And because we assumed a path here, there weren't 00:59:35.080 --> 00:59:37.330 align:middle line:90% any duplicate edges within x. 00:59:37.330 --> 00:59:39.590 align:middle line:90% And so I think we're good. 00:59:39.590 --> 00:59:42.816 align:middle line:90% And that proves a contradiction. 00:59:42.816 --> 00:59:45.790 align:middle line:90% 00:59:45.790 --> 00:59:49.910 align:middle line:84% Because we assumed that T was the longest trail, therefore, 00:59:49.910 --> 00:59:53.240 align:middle line:84% T actually visits every edge in the graph. 00:59:53.240 --> 00:59:55.330 align:middle line:84% It also visits every vertex at least 00:59:55.330 --> 00:59:57.380 align:middle line:84% once, just because the graph is connected. 00:59:57.380 --> 01:00:00.050 align:middle line:84% If you visit all the edges and your graph is connected, 01:00:00.050 --> 01:00:02.540 align:middle line:84% then you definitely visit all the vertices. 01:00:02.540 --> 01:00:06.290 align:middle line:90% So that proves this theorem. 01:00:06.290 --> 01:00:07.630 align:middle line:90% Any questions? 01:00:07.630 --> 01:00:09.880 align:middle line:90% A lot of details there. 01:00:09.880 --> 01:00:12.255 align:middle line:84% That's one of our bigger proofs, but it's pretty cool. 01:00:12.255 --> 01:00:18.280 align:middle line:90% 01:00:18.280 --> 01:00:22.400 align:middle line:90% Let me-- this was Euler tours. 01:00:22.400 --> 01:00:26.620 align:middle line:84% You might also wonder about Euler walks. 01:00:26.620 --> 01:00:29.680 align:middle line:84% So we've now shown in particular that this graph 01:00:29.680 --> 01:00:32.540 align:middle line:84% with the extra vertex here is indeed Eulerian. 01:00:32.540 --> 01:00:36.540 align:middle line:84% I mean, I also drew it, but we now understand why. 01:00:36.540 --> 01:00:39.180 align:middle line:84% When I remove this vertex, not all the vertex degrees 01:00:39.180 --> 01:00:40.320 align:middle line:90% were even. 01:00:40.320 --> 01:00:42.420 align:middle line:90% Two of them were odd. 01:00:42.420 --> 01:00:45.870 align:middle line:84% It turns out that's the right answer. 01:00:45.870 --> 01:00:50.040 align:middle line:84% So corollary, remember, means a theorem 01:00:50.040 --> 01:00:52.170 align:middle line:84% you can prove from the previous theorem, 01:00:52.170 --> 01:00:54.600 align:middle line:90% or follows fairly easily. 01:00:54.600 --> 01:01:02.810 align:middle line:84% So I claim that a graph has an open Euler walk. 01:01:02.810 --> 01:01:06.570 align:middle line:90% 01:01:06.570 --> 01:01:08.533 align:middle line:90% Open here just means not closed. 01:01:08.533 --> 01:01:11.070 align:middle line:90% 01:01:11.070 --> 01:01:13.170 align:middle line:84% So we've already characterized the situation 01:01:13.170 --> 01:01:14.470 align:middle line:90% of closed Euler walks. 01:01:14.470 --> 01:01:15.900 align:middle line:90% That's this theorem. 01:01:15.900 --> 01:01:17.430 align:middle line:90% That's the Euler tour. 01:01:17.430 --> 01:01:18.847 align:middle line:84% So what's left is to characterize 01:01:18.847 --> 01:01:21.180 align:middle line:84% when there's an open Euler walk, where you start and end 01:01:21.180 --> 01:01:22.690 align:middle line:90% at different vertices. 01:01:22.690 --> 01:01:25.440 align:middle line:84% So let's ignore-- let's suppose that we 01:01:25.440 --> 01:01:29.050 align:middle line:84% don't have a closed Euler walk, but we do have an open one. 01:01:29.050 --> 01:01:30.780 align:middle line:84% I claim this is possible if and only 01:01:30.780 --> 01:01:36.490 align:middle line:84% if the graph is connected as before, and exactly 01:01:36.490 --> 01:01:39.510 align:middle line:90% two vertices have odd degree. 01:01:39.510 --> 01:01:48.820 align:middle line:90% 01:01:48.820 --> 01:01:54.460 align:middle line:84% Intuitively, those odd degree vertices are where you start 01:01:54.460 --> 01:01:55.790 align:middle line:90% and where you end. 01:01:55.790 --> 01:01:59.560 align:middle line:84% If you look at this example with this erased, 01:01:59.560 --> 01:02:02.360 align:middle line:84% I started here, which had degree 3. 01:02:02.360 --> 01:02:06.400 align:middle line:84% I walked around, and I ended here, which also had degree 3. 01:02:06.400 --> 01:02:09.760 align:middle line:84% So the odd degree is essentially to compensate for starting 01:02:09.760 --> 01:02:11.390 align:middle line:90% and ending at those vertices. 01:02:11.390 --> 01:02:13.490 align:middle line:84% And that turns out to be all you can do. 01:02:13.490 --> 01:02:15.800 align:middle line:84% So if you have a graph with all even degree vertices, 01:02:15.800 --> 01:02:17.360 align:middle line:90% then there's an Euler tour. 01:02:17.360 --> 01:02:21.070 align:middle line:84% And if you have a graph with two odd degree vertices, 01:02:21.070 --> 01:02:22.460 align:middle line:90% then there's an Euler walk. 01:02:22.460 --> 01:02:25.700 align:middle line:84% If you have more than two odd degree vertices, 01:02:25.700 --> 01:02:27.650 align:middle line:90% like here, there's neither. 01:02:27.650 --> 01:02:31.373 align:middle line:84% What if I have one odd degree vertex? 01:02:31.373 --> 01:02:35.220 align:middle line:90% 01:02:35.220 --> 01:02:36.000 align:middle line:90% Yeah. 01:02:36.000 --> 01:02:36.850 align:middle line:90% AUDIENCE: That's impossible. 01:02:36.850 --> 01:02:38.200 align:middle line:90% ERIK DEMAINE: That's impossible. 01:02:38.200 --> 01:02:39.325 align:middle line:90% What are you talking about? 01:02:39.325 --> 01:02:41.590 align:middle line:84% Because the number of odd degree vertices is even. 01:02:41.590 --> 01:02:45.180 align:middle line:84% I think you probably showed that in recitation, or even lecture. 01:02:45.180 --> 01:02:45.870 align:middle line:90% I forget. 01:02:45.870 --> 01:02:48.630 align:middle line:84% It follows from the handshaking lemma from lecture. 01:02:48.630 --> 01:02:51.120 align:middle line:90% So those are your options. 01:02:51.120 --> 01:02:52.740 align:middle line:90% Cool. 01:02:52.740 --> 01:02:54.640 align:middle line:90% Let me sketch a proof of this. 01:02:54.640 --> 01:02:59.353 align:middle line:84% And then I want to go to the last topic, which is trees. 01:02:59.353 --> 01:03:04.110 align:middle line:90% 01:03:04.110 --> 01:03:10.060 align:middle line:84% So I want to-- the easy direction, as before, 01:03:10.060 --> 01:03:12.490 align:middle line:84% if you have an Euler walk, you're still connected. 01:03:12.490 --> 01:03:14.460 align:middle line:84% And if you do the counting right, 01:03:14.460 --> 01:03:19.920 align:middle line:84% you can see the number of degree 2 vertices should be odd, 01:03:19.920 --> 01:03:23.910 align:middle line:84% because it's just like this proof. 01:03:23.910 --> 01:03:25.870 align:middle line:90% But now it's not symmetric. 01:03:25.870 --> 01:03:28.970 align:middle line:84% We can no longer do these things modulo k, which was really nice. 01:03:28.970 --> 01:03:31.750 align:middle line:84% Every visit looked the same, because now there's-- 01:03:31.750 --> 01:03:34.030 align:middle line:84% v0 and vk are visited in a different way 01:03:34.030 --> 01:03:35.600 align:middle line:90% from every other vertex. 01:03:35.600 --> 01:03:39.910 align:middle line:84% And just like we said before, the last vertex, the last visit 01:03:39.910 --> 01:03:42.200 align:middle line:84% has only one edge coming into it. 01:03:42.200 --> 01:03:43.688 align:middle line:90% So that makes this one odd. 01:03:43.688 --> 01:03:45.980 align:middle line:84% And the first vertex has only one edge going out of it, 01:03:45.980 --> 01:03:47.180 align:middle line:90% so it makes that vertex odd. 01:03:47.180 --> 01:03:50.710 align:middle line:90% So this is easy. 01:03:50.710 --> 01:03:53.770 align:middle line:84% The proof is in the notes if you're interested. 01:03:53.770 --> 01:03:57.190 align:middle line:84% The interesting part is the other direction. 01:03:57.190 --> 01:04:00.033 align:middle line:84% I mean, this is roughly as before. 01:04:00.033 --> 01:04:03.760 align:middle line:90% 01:04:03.760 --> 01:04:05.620 align:middle line:84% Let's suppose that we have exactly two 01:04:05.620 --> 01:04:06.980 align:middle line:90% vertices of odd degree. 01:04:06.980 --> 01:04:09.020 align:middle line:84% I want to claim there's an open Euler walk. 01:04:09.020 --> 01:04:12.470 align:middle line:84% And I want to do it without repeating this giant proof. 01:04:12.470 --> 01:04:13.980 align:middle line:90% That's a lot of work to-- 01:04:13.980 --> 01:04:15.980 align:middle line:84% I mean, you could repeat it and be more careful. 01:04:15.980 --> 01:04:17.605 align:middle line:84% And you can prove the Euler walk thing. 01:04:17.605 --> 01:04:20.080 align:middle line:84% But in fact, there's a way to convert this problem 01:04:20.080 --> 01:04:22.400 align:middle line:90% into the Euler tour problem. 01:04:22.400 --> 01:04:23.900 align:middle line:84% So I'm going to do it by picture. 01:04:23.900 --> 01:04:25.630 align:middle line:90% Suppose you have a graph. 01:04:25.630 --> 01:04:32.190 align:middle line:84% And you have two vertices, u and v, which are odd degree. 01:04:32.190 --> 01:04:35.910 align:middle line:84% Maybe they're also just odd in general, but their odd degree. 01:04:35.910 --> 01:04:38.950 align:middle line:84% And now I'm going to modify the graph. 01:04:38.950 --> 01:04:41.770 align:middle line:84% I'm going to construct a new graph, G prime. 01:04:41.770 --> 01:04:46.030 align:middle line:84% So this graph that I drew is G here. 01:04:46.030 --> 01:04:53.850 align:middle line:84% And in G prime, I'm going to add a new vertex x and two edges, 01:04:53.850 --> 01:05:01.440 align:middle line:84% from u to x and x to v. Why did I do that? 01:05:01.440 --> 01:05:04.680 align:middle line:84% Because now all the degrees are even. 01:05:04.680 --> 01:05:07.870 align:middle line:90% In G, the degree of u was odd. 01:05:07.870 --> 01:05:10.120 align:middle line:90% And in G prime, it's one larger. 01:05:10.120 --> 01:05:11.140 align:middle line:90% So now it's even. 01:05:11.140 --> 01:05:13.780 align:middle line:90% Same for v. And x has degree 2. 01:05:13.780 --> 01:05:15.490 align:middle line:90% So now everything's even. 01:05:15.490 --> 01:05:23.040 align:middle line:84% So that means G prime has an Euler tour 01:05:23.040 --> 01:05:25.330 align:middle line:90% by the previous theorem. 01:05:25.330 --> 01:05:27.910 align:middle line:84% So that was-- I don't have to repeat the proof. 01:05:27.910 --> 01:05:30.310 align:middle line:84% I can apply the theorem as a whole. 01:05:30.310 --> 01:05:44.470 align:middle line:84% And now I want to convert this tour into an Euler walk in G. 01:05:44.470 --> 01:05:46.130 align:middle line:90% How do I do that? 01:05:46.130 --> 01:05:49.870 align:middle line:90% I just erase these two edges. 01:05:49.870 --> 01:05:54.970 align:middle line:84% The Euler tour visits every edge in G prime exactly once. 01:05:54.970 --> 01:05:56.150 align:middle line:90% And it's a loop. 01:05:56.150 --> 01:05:57.760 align:middle line:84% So in particular, at some point, it 01:05:57.760 --> 01:06:00.080 align:middle line:84% crosses over, let's say, from u to x. 01:06:00.080 --> 01:06:02.110 align:middle line:84% And then it can't go back along the same edge, 01:06:02.110 --> 01:06:04.280 align:middle line:90% because you can't repeat edges. 01:06:04.280 --> 01:06:06.980 align:middle line:84% So then it has to go-- let's say it goes from u to x to v, 01:06:06.980 --> 01:06:09.550 align:middle line:84% or it might go from v to x to u at some point 01:06:09.550 --> 01:06:15.910 align:middle line:90% in the closed walk, in the tour. 01:06:15.910 --> 01:06:20.620 align:middle line:84% And when it does that, I want to-- 01:06:20.620 --> 01:06:24.400 align:middle line:84% instead of going from somewhere to u to x to v, 01:06:24.400 --> 01:06:26.430 align:middle line:84% I'm going to start at v. And then I'm 01:06:26.430 --> 01:06:31.290 align:middle line:84% going to keep going along the tour until I get back to u. 01:06:31.290 --> 01:06:34.170 align:middle line:84% And that will include everything in the tour, 01:06:34.170 --> 01:06:36.910 align:middle line:90% except for these two edges. 01:06:36.910 --> 01:06:39.910 align:middle line:84% And so that, actually, will be an Euler walk in G. 01:06:39.910 --> 01:06:43.470 align:middle line:84% So that maybe went a little fast there, but hopefully 01:06:43.470 --> 01:06:45.040 align:middle line:90% that's fairly intuitive. 01:06:45.040 --> 01:06:48.270 align:middle line:84% This is a very powerful idea called reduction, 01:06:48.270 --> 01:06:52.115 align:middle line:84% where you take your problem that you want to solve 01:06:52.115 --> 01:06:53.490 align:middle line:84% and you convert it into a problem 01:06:53.490 --> 01:06:55.090 align:middle line:84% that you already know how to solve. 01:06:55.090 --> 01:06:57.940 align:middle line:84% So we already-- we just solved Euler tours. 01:06:57.940 --> 01:07:00.460 align:middle line:84% And now we want to solve open Euler walks. 01:07:00.460 --> 01:07:02.700 align:middle line:90% And we could do it by adding-- 01:07:02.700 --> 01:07:05.895 align:middle line:84% augmenting the graph a little bit, applying the theorem, 01:07:05.895 --> 01:07:07.770 align:middle line:84% and then we need to make sure that the result 01:07:07.770 --> 01:07:10.228 align:middle line:84% from that theorem is actually good enough for what we need. 01:07:10.228 --> 01:07:13.830 align:middle line:84% We can convert our Euler tour into an Euler walk. 01:07:13.830 --> 01:07:16.660 align:middle line:84% But because tours are cyclic in nature, 01:07:16.660 --> 01:07:19.530 align:middle line:84% you can rotate it to start at v and keep going 01:07:19.530 --> 01:07:22.200 align:middle line:84% and visit everything and end at u. 01:07:22.200 --> 01:07:22.700 align:middle line:90% Cool. 01:07:22.700 --> 01:07:24.830 align:middle line:84% So that also proves that you start and end 01:07:24.830 --> 01:07:29.570 align:middle line:84% at the two odd degree vertices in G, which is nice. 01:07:29.570 --> 01:07:30.560 align:middle line:90% Questions? 01:07:30.560 --> 01:07:35.030 align:middle line:90% 01:07:35.030 --> 01:07:36.543 align:middle line:90% Let's go back to-- 01:07:36.543 --> 01:07:42.770 align:middle line:90% 01:07:42.770 --> 01:07:45.920 align:middle line:90% back to trees. 01:07:45.920 --> 01:07:49.760 align:middle line:84% So we talked about walks, paths, cycles, connectivity, 01:07:49.760 --> 01:07:51.440 align:middle line:90% Euler things. 01:07:51.440 --> 01:07:53.635 align:middle line:90% And the last topic is trees. 01:07:53.635 --> 01:08:02.270 align:middle line:90% 01:08:02.270 --> 01:08:03.580 align:middle line:90% That's a very sad board. 01:08:03.580 --> 01:08:17.870 align:middle line:90% 01:08:17.870 --> 01:08:20.450 align:middle line:84% I guess now for something completely different. 01:08:20.450 --> 01:08:24.886 align:middle line:84% Now we know how to draw graphs without raising our pen. 01:08:24.886 --> 01:08:28.140 align:middle line:84% I define something called a tree, which you've probably 01:08:28.140 --> 01:08:31.560 align:middle line:84% heard the word tree many times in the past, not just 01:08:31.560 --> 01:08:34.590 align:middle line:84% nature trees, but also computer science trees 01:08:34.590 --> 01:08:37.920 align:middle line:84% are all over the place, like your directory tree. 01:08:37.920 --> 01:08:38.604 align:middle line:90% I don't know. 01:08:38.604 --> 01:08:39.729 align:middle line:90% There are trees everywhere. 01:08:39.729 --> 01:08:41.550 align:middle line:84% If you've done any data structures, 01:08:41.550 --> 01:08:44.136 align:middle line:84% you know about trees, tree data structures. 01:08:44.136 --> 01:08:47.399 align:middle line:90% 01:08:47.399 --> 01:08:50.090 align:middle line:84% So what is the mathematical notion of a tree? 01:08:50.090 --> 01:08:57.390 align:middle line:90% 01:08:57.390 --> 01:08:58.479 align:middle line:90% It's a graph. 01:08:58.479 --> 01:09:02.529 align:middle line:84% And again, today, we're using simple undirected graphs. 01:09:02.529 --> 01:09:04.439 align:middle line:90% And trees always require that. 01:09:04.439 --> 01:09:07.720 align:middle line:84% It's a graph that is both connected and acyclic. 01:09:07.720 --> 01:09:12.149 align:middle line:90% Acyclic just means no cycles. 01:09:12.149 --> 01:09:13.510 align:middle line:90% What's a cycle? 01:09:13.510 --> 01:09:14.380 align:middle line:90% It's been a while. 01:09:14.380 --> 01:09:18.420 align:middle line:84% Remember, a cycle is a closed path of positive length. 01:09:18.420 --> 01:09:20.859 align:middle line:84% So it doesn't repeat any vertices or edges, 01:09:20.859 --> 01:09:25.560 align:middle line:84% except that the very first vertex equals the last vertex. 01:09:25.560 --> 01:09:29.859 align:middle line:84% So if there are no cycles, there's no way to do this. 01:09:29.859 --> 01:09:31.300 align:middle line:90% Then we call it a tree. 01:09:31.300 --> 01:09:32.803 align:middle line:90% And some examples of trees. 01:09:32.803 --> 01:09:37.329 align:middle line:90% 01:09:37.329 --> 01:09:37.829 align:middle line:90% Sure. 01:09:37.829 --> 01:09:40.470 align:middle line:90% 01:09:40.470 --> 01:09:41.735 align:middle line:90% Just a path is a tree. 01:09:41.735 --> 01:09:45.660 align:middle line:90% 01:09:45.660 --> 01:09:48.720 align:middle line:84% This is a special kind of tree we call a star-- 01:09:48.720 --> 01:09:50.580 align:middle line:84% one vertex connected to a bunch of things. 01:09:50.580 --> 01:09:52.510 align:middle line:84% In general, it looks something like this. 01:09:52.510 --> 01:09:54.279 align:middle line:84% If you've seen computer science trees, 01:09:54.279 --> 01:09:56.670 align:middle line:84% this can be a little bit different. 01:09:56.670 --> 01:09:58.567 align:middle line:84% So the only thing we're requiring here 01:09:58.567 --> 01:09:59.650 align:middle line:90% is that they're connected. 01:09:59.650 --> 01:10:00.400 align:middle line:90% So this is a tree. 01:10:00.400 --> 01:10:01.025 align:middle line:90% This is a tree. 01:10:01.025 --> 01:10:02.440 align:middle line:84% But together, this is not a tree. 01:10:02.440 --> 01:10:04.180 align:middle line:84% It's actually what we call a forest. 01:10:04.180 --> 01:10:06.480 align:middle line:84% This would be a forest with three trees in it. 01:10:06.480 --> 01:10:08.795 align:middle line:84% Or individually, these are each a tree graph. 01:10:08.795 --> 01:10:14.190 align:middle line:90% 01:10:14.190 --> 01:10:16.890 align:middle line:84% Let's prove some properties about trees 01:10:16.890 --> 01:10:18.050 align:middle line:90% because they're so useful. 01:10:18.050 --> 01:10:26.040 align:middle line:90% 01:10:26.040 --> 01:10:27.260 align:middle line:90% Just nice to work with. 01:10:27.260 --> 01:10:30.900 align:middle line:90% 01:10:30.900 --> 01:10:32.760 align:middle line:84% In particular, there's a very useful notion 01:10:32.760 --> 01:10:36.030 align:middle line:90% about trees called leaves. 01:10:36.030 --> 01:10:43.690 align:middle line:84% You can guess where this came from, from real trees. 01:10:43.690 --> 01:10:47.440 align:middle line:90% But the idea is very simple. 01:10:47.440 --> 01:10:49.210 align:middle line:84% Just a leaf is a vertex of degree 1. 01:10:49.210 --> 01:10:52.450 align:middle line:90% You can apply this to any graph. 01:10:52.450 --> 01:10:54.040 align:middle line:84% Sometimes you have vertices degree 1. 01:10:54.040 --> 01:10:55.240 align:middle line:90% Sometimes you don't. 01:10:55.240 --> 01:10:58.050 align:middle line:90% But trees have leaves. 01:10:58.050 --> 01:11:00.510 align:middle line:90% Shocking news. 01:11:00.510 --> 01:11:01.860 align:middle line:90% So this is-- 01:11:01.860 --> 01:11:04.320 align:middle line:90% I'll call it the leaf lemma. 01:11:04.320 --> 01:11:08.170 align:middle line:84% It's not quite true that all trees have leaves. 01:11:08.170 --> 01:11:13.170 align:middle line:84% They have to start growing for a little while first. 01:11:13.170 --> 01:11:17.290 align:middle line:84% And in mathematics, this is every tree of-- 01:11:17.290 --> 01:11:20.240 align:middle line:90% 01:11:20.240 --> 01:11:22.790 align:middle line:90% I guess width is better-- 01:11:22.790 --> 01:11:34.040 align:middle line:84% at least two vertices has at least two leaves. 01:11:34.040 --> 01:11:36.360 align:middle line:90% So as long as your graph-- 01:11:36.360 --> 01:11:38.822 align:middle line:84% as long as your tree is not this graph-- 01:11:38.822 --> 01:11:40.530 align:middle line:84% maybe I'll add it to my list of examples. 01:11:40.530 --> 01:11:41.970 align:middle line:90% These were all trees. 01:11:41.970 --> 01:11:47.420 align:middle line:84% Also, single vertex is a tree, but it doesn't have any edges. 01:11:47.420 --> 01:11:50.910 align:middle line:84% And it has degree 0 vertex, no degree 1 vertices. 01:11:50.910 --> 01:11:53.250 align:middle line:84% So other than this exceptional case, 01:11:53.250 --> 01:11:56.060 align:middle line:84% every other tree has at least two leaves. 01:11:56.060 --> 01:11:57.660 align:middle line:84% Where are the leaves in these graphs? 01:11:57.660 --> 01:11:58.830 align:middle line:90% Here's a leaf. 01:11:58.830 --> 01:11:59.657 align:middle line:90% Here's a leaf. 01:11:59.657 --> 01:12:00.240 align:middle line:90% Here's a leaf. 01:12:00.240 --> 01:12:01.760 align:middle line:90% Here's a leaf. 01:12:01.760 --> 01:12:03.208 align:middle line:90% This one just has two. 01:12:03.208 --> 01:12:04.750 align:middle line:84% This one has a whole bunch of leaves. 01:12:04.750 --> 01:12:07.910 align:middle line:90% 01:12:07.910 --> 01:12:16.230 align:middle line:84% Let's sketch a proof of this lemma, 01:12:16.230 --> 01:12:19.690 align:middle line:90% which is a familiar technique. 01:12:19.690 --> 01:12:22.840 align:middle line:90% Take a longest something. 01:12:22.840 --> 01:12:25.380 align:middle line:84% In this case, the something is path. 01:12:25.380 --> 01:12:27.930 align:middle line:84% Remember, a path is a walk that doesn't repeat any vertices 01:12:27.930 --> 01:12:29.020 align:middle line:90% or edges. 01:12:29.020 --> 01:12:31.510 align:middle line:84% So I want to take a longest path in one of these graphs. 01:12:31.510 --> 01:12:36.240 align:middle line:84% For example, if I take this, this 01:12:36.240 --> 01:12:40.740 align:middle line:84% turns out to be a longest path in this graph, length 4. 01:12:40.740 --> 01:12:43.950 align:middle line:84% I believe you cannot find a path of length 5 here. 01:12:43.950 --> 01:12:47.250 align:middle line:84% You can find walks that are arbitrarily long, but paths, 01:12:47.250 --> 01:12:49.750 align:middle line:84% you can only find length 4 in this particular graph. 01:12:49.750 --> 01:12:54.600 align:middle line:84% I claim that the first vertex and the last vertex in that path 01:12:54.600 --> 01:12:58.050 align:middle line:90% must be leaves. 01:12:58.050 --> 01:13:01.583 align:middle line:90% So ends must be leaves. 01:13:01.583 --> 01:13:05.190 align:middle line:90% 01:13:05.190 --> 01:13:06.433 align:middle line:90% And they must be distinct. 01:13:06.433 --> 01:13:10.560 align:middle line:90% 01:13:10.560 --> 01:13:12.130 align:middle line:90% Why are they distinct? 01:13:12.130 --> 01:13:14.020 align:middle line:90% Well, this is a path. 01:13:14.020 --> 01:13:15.660 align:middle line:84% And a path can't repeat vertices. 01:13:15.660 --> 01:13:18.260 align:middle line:84% So unless the path has length zero, 01:13:18.260 --> 01:13:21.660 align:middle line:84% the start vertex and the end vertex are different. 01:13:21.660 --> 01:13:24.748 align:middle line:84% It's not length zero, basically, because we're not in this case. 01:13:24.748 --> 01:13:26.040 align:middle line:90% In this case, there's no edges. 01:13:26.040 --> 01:13:27.180 align:middle line:90% You can't follow anything. 01:13:27.180 --> 01:13:28.530 align:middle line:84% But as long as there's at least one edge, 01:13:28.530 --> 01:13:30.738 align:middle line:84% then you know there is a path of length at least one. 01:13:30.738 --> 01:13:34.145 align:middle line:84% And so the start vertex and the end vertex will be different. 01:13:34.145 --> 01:13:36.020 align:middle line:84% And furthermore, I claim they must be leaves. 01:13:36.020 --> 01:13:38.340 align:middle line:90% Basically, if this wasn't-- 01:13:38.340 --> 01:13:41.330 align:middle line:84% if this vertex didn't have degree 1, so there wasn't just 01:13:41.330 --> 01:13:44.870 align:middle line:84% this one way to exit, there'd be another edge incident 01:13:44.870 --> 01:13:45.600 align:middle line:90% to that vertex. 01:13:45.600 --> 01:13:50.210 align:middle line:84% And then you could have made a longer path, so a contradiction. 01:13:50.210 --> 01:13:51.840 align:middle line:90% I'm going to call it there. 01:13:51.840 --> 01:13:53.763 align:middle line:84% And you can read the notes for more details. 01:13:53.763 --> 01:13:57.560 align:middle line:90% 01:13:57.560 --> 01:14:00.050 align:middle line:84% And instead, I'm going to tell you what 01:14:00.050 --> 01:14:01.260 align:middle line:90% you can do with these leaves. 01:14:01.260 --> 01:14:02.093 align:middle line:90% Why are they useful? 01:14:02.093 --> 01:14:06.260 align:middle line:90% 01:14:06.260 --> 01:14:10.630 align:middle line:84% Every tree except the one vertex tree has a leaf. 01:14:10.630 --> 01:14:13.240 align:middle line:90% 01:14:13.240 --> 01:14:15.730 align:middle line:84% And if you've ever taken care of a plant, 01:14:15.730 --> 01:14:18.010 align:middle line:84% sometimes it's useful to prune the plant. 01:14:18.010 --> 01:14:19.540 align:middle line:90% It grows better. 01:14:19.540 --> 01:14:26.380 align:middle line:84% In our case, pruning is a very useful operation 01:14:26.380 --> 01:14:29.950 align:middle line:90% that lets us use induction. 01:14:29.950 --> 01:14:33.680 align:middle line:84% We've seen some inductions for graphs. 01:14:33.680 --> 01:14:36.610 align:middle line:84% And trees are really nice to induct on, because they 01:14:36.610 --> 01:14:38.373 align:middle line:90% have leaves to trim. 01:14:38.373 --> 01:14:41.140 align:middle line:90% 01:14:41.140 --> 01:14:50.410 align:middle line:84% So this is a theorem for any leaf l in a tree 01:14:50.410 --> 01:14:55.360 align:middle line:90% T, T minus l is a tree. 01:14:55.360 --> 01:14:57.950 align:middle line:90% 01:14:57.950 --> 01:14:59.660 align:middle line:90% What is T minus l? 01:14:59.660 --> 01:15:12.110 align:middle line:84% This is the graph induced by all the vertices minus l. 01:15:12.110 --> 01:15:13.610 align:middle line:84% So in other words, it's the graph 01:15:13.610 --> 01:15:16.190 align:middle line:84% you get if you take the tree and you clip off 01:15:16.190 --> 01:15:20.013 align:middle line:90% the leaf and its incident edge. 01:15:20.013 --> 01:15:21.180 align:middle line:90% That's the negative version. 01:15:21.180 --> 01:15:25.730 align:middle line:84% The positive version is you say, I keep everything except l. 01:15:25.730 --> 01:15:28.340 align:middle line:84% And I take all the edges among those vertices 01:15:28.340 --> 01:15:29.300 align:middle line:90% that do not include l. 01:15:29.300 --> 01:15:31.500 align:middle line:90% That will end up having-- 01:15:31.500 --> 01:15:33.860 align:middle line:90% let's call this T prime. 01:15:33.860 --> 01:15:36.440 align:middle line:84% The number of edges in T prime is 01:15:36.440 --> 01:15:40.560 align:middle line:84% going to be 1 less than the number of edges in T. 01:15:40.560 --> 01:15:42.320 align:middle line:84% And the number of vertices is also 01:15:42.320 --> 01:15:47.340 align:middle line:84% going to be one less, because we removed one vertex and one edge. 01:15:47.340 --> 01:15:48.710 align:middle line:90% So here's our tree. 01:15:48.710 --> 01:15:51.320 align:middle line:90% Here's a leaf. 01:15:51.320 --> 01:15:57.300 align:middle line:84% This whole picture is T. And this inner picture is T prime. 01:15:57.300 --> 01:15:59.240 align:middle line:90% This is a leaf l. 01:15:59.240 --> 01:16:04.745 align:middle line:84% So T prime or T minus l is this smaller thing. 01:16:04.745 --> 01:16:06.120 align:middle line:84% A very useful thing-- in general, 01:16:06.120 --> 01:16:07.670 align:middle line:84% if you remove a vertex from a graph, 01:16:07.670 --> 01:16:10.910 align:middle line:84% you also remove all of its incident edges. 01:16:10.910 --> 01:16:14.830 align:middle line:84% So this is a thing which is pretty easy to prove, 01:16:14.830 --> 01:16:17.110 align:middle line:84% because what we're saying is suppose that T 01:16:17.110 --> 01:16:19.570 align:middle line:90% is connected and acyclic. 01:16:19.570 --> 01:16:22.400 align:middle line:84% The claim is that T prime is also connected in acyclic. 01:16:22.400 --> 01:16:25.910 align:middle line:84% And you can just show that that's true. 01:16:25.910 --> 01:16:28.300 align:middle line:90% It's not very hard. 01:16:28.300 --> 01:16:30.640 align:middle line:90% Again, see the notes. 01:16:30.640 --> 01:16:32.710 align:middle line:84% And what this enables to do, together, 01:16:32.710 --> 01:16:35.050 align:middle line:84% the leaf lemma and this tree-pruning lemma 01:16:35.050 --> 01:16:38.680 align:middle line:84% let you do induction over trees in a really nice way. 01:16:38.680 --> 01:16:43.030 align:middle line:84% So let's argue-- let's see an example of that with something 01:16:43.030 --> 01:16:46.390 align:middle line:90% called the size of the tree. 01:16:46.390 --> 01:17:00.100 align:middle line:84% Any tree with n vertices has n minus 1 edges always. 01:17:00.100 --> 01:17:03.820 align:middle line:84% If you look at these examples, the one vertex tree 01:17:03.820 --> 01:17:05.570 align:middle line:90% has zero edges. 01:17:05.570 --> 01:17:09.060 align:middle line:84% The four vertex path has three edges. 01:17:09.060 --> 01:17:12.960 align:middle line:84% The six vertex star has five edges. 01:17:12.960 --> 01:17:16.340 align:middle line:84% This thing has one fewer edge than vertices. 01:17:16.340 --> 01:17:17.728 align:middle line:90% I won't count. 01:17:17.728 --> 01:17:20.270 align:middle line:84% I guess there are three kinds of mathematicians in the world. 01:17:20.270 --> 01:17:23.940 align:middle line:84% Those who can count, and those who can't. 01:17:23.940 --> 01:17:26.440 align:middle line:90% So I'm the latter kind. 01:17:26.440 --> 01:17:29.120 align:middle line:90% 01:17:29.120 --> 01:17:31.250 align:middle line:84% So finally, we get to an induction. 01:17:31.250 --> 01:17:32.940 align:middle line:84% We haven't done induction all day. 01:17:32.940 --> 01:17:34.040 align:middle line:90% It's terrible. 01:17:34.040 --> 01:17:37.010 align:middle line:90% Finally, we can do induction. 01:17:37.010 --> 01:17:39.293 align:middle line:84% So we're going to-- remember, whenever 01:17:39.293 --> 01:17:41.210 align:middle line:84% you do an induction over graphs, you're really 01:17:41.210 --> 01:17:44.592 align:middle line:84% just doing regular induction or strong induction over numbers. 01:17:44.592 --> 01:17:46.550 align:middle line:84% And so in this case, we're going to induct over 01:17:46.550 --> 01:17:48.360 align:middle line:84% the number of vertices in the graph. 01:17:48.360 --> 01:17:51.830 align:middle line:84% So our induction hypothesis is this, that for a number n, 01:17:51.830 --> 01:17:57.500 align:middle line:84% P of n is the statement that any tree with n vertices 01:17:57.500 --> 01:18:00.110 align:middle line:90% has exactly n minus 1 edges. 01:18:00.110 --> 01:18:01.390 align:middle line:90% I should write exactly here. 01:18:01.390 --> 01:18:04.472 align:middle line:90% 01:18:04.472 --> 01:18:06.180 align:middle line:84% And we're going to prove it by induction. 01:18:06.180 --> 01:18:09.070 align:middle line:90% So we probably need a base case. 01:18:09.070 --> 01:18:12.430 align:middle line:84% What is the smallest possible thing that n can be? 01:18:12.430 --> 01:18:13.520 align:middle line:90% 1. 01:18:13.520 --> 01:18:16.610 align:middle line:84% Remember, we forbid zero vertex graphs. 01:18:16.610 --> 01:18:18.830 align:middle line:84% You have to have at least one vertex to be a graph. 01:18:18.830 --> 01:18:22.930 align:middle line:84% So in n equals 1, that means there's only one graph with n 01:18:22.930 --> 01:18:25.400 align:middle line:90% equals 1, and it has zero edges. 01:18:25.400 --> 01:18:28.390 align:middle line:90% Great. 01:18:28.390 --> 01:18:31.255 align:middle line:84% Well, my vertex looks a lot like my number 0. 01:18:31.255 --> 01:18:31.755 align:middle line:90% Sorry. 01:18:31.755 --> 01:18:34.400 align:middle line:90% 01:18:34.400 --> 01:18:36.350 align:middle line:90% So let's do the induction step. 01:18:36.350 --> 01:18:38.770 align:middle line:84% Now we're going to use all of these beautiful lemmas 01:18:38.770 --> 01:18:42.100 align:middle line:90% that we have about leaves. 01:18:42.100 --> 01:18:46.010 align:middle line:84% So let's suppose that n is bigger than 1. 01:18:46.010 --> 01:18:48.400 align:middle line:84% In other words, n is at least two. 01:18:48.400 --> 01:18:50.540 align:middle line:84% And therefore, the leaf lemma applies. 01:18:50.540 --> 01:18:52.190 align:middle line:84% It says if you have n at least two, 01:18:52.190 --> 01:18:54.110 align:middle line:90% you have at least two leaves. 01:18:54.110 --> 01:18:56.350 align:middle line:84% I actually only need one of them. 01:18:56.350 --> 01:19:05.150 align:middle line:84% But let's say, by leaf lemma, there exists a leaf. 01:19:05.150 --> 01:19:07.700 align:middle line:90% And let's call it l. 01:19:07.700 --> 01:19:12.080 align:middle line:84% Now, I'm going to apply this tree pruning lemma that 01:19:12.080 --> 01:19:13.550 align:middle line:90% says now I have a leaf. 01:19:13.550 --> 01:19:15.590 align:middle line:90% I can remove it. 01:19:15.590 --> 01:19:16.590 align:middle line:90% Clip it. 01:19:16.590 --> 01:19:23.370 align:middle line:84% Take a new graph, T prime, which is everything in T except l. 01:19:23.370 --> 01:19:34.315 align:middle line:84% So by tree pruning T prime, which is T minus l, is a tree. 01:19:34.315 --> 01:19:37.430 align:middle line:90% 01:19:37.430 --> 01:19:38.700 align:middle line:90% Not only is it a tree. 01:19:38.700 --> 01:19:40.030 align:middle line:90% It's a smaller tree. 01:19:40.030 --> 01:19:42.920 align:middle line:90% 01:19:42.920 --> 01:19:46.203 align:middle line:84% It has exactly n minus 1 vertices. 01:19:46.203 --> 01:19:49.580 align:middle line:90% 01:19:49.580 --> 01:19:55.280 align:middle line:84% So I can apply induction, because in their induction step, 01:19:55.280 --> 01:19:58.570 align:middle line:84% we get to assume that this property is true for all smaller 01:19:58.570 --> 01:19:59.070 align:middle line:90% n. 01:19:59.070 --> 01:20:00.120 align:middle line:90% That's strong induction. 01:20:00.120 --> 01:20:01.860 align:middle line:84% In particular, it's true for n minus 1. 01:20:01.860 --> 01:20:03.970 align:middle line:90% That's weak induction. 01:20:03.970 --> 01:20:08.320 align:middle line:84% So by the induction hypothesis, we 01:20:08.320 --> 01:20:16.090 align:middle line:84% know that T prime has one fewer edge than vertex, 01:20:16.090 --> 01:20:19.100 align:middle line:90% and it has n minus 1 vertices. 01:20:19.100 --> 01:20:21.973 align:middle line:90% And so it has n minus 2 edges. 01:20:21.973 --> 01:20:25.480 align:middle line:90% 01:20:25.480 --> 01:20:29.300 align:middle line:84% And T prime was everything in T except for one edge, 01:20:29.300 --> 01:20:32.200 align:middle line:90% that measly edge connected to l. 01:20:32.200 --> 01:20:39.495 align:middle line:84% So T has one more edge than T prime. 01:20:39.495 --> 01:20:44.500 align:middle line:90% 01:20:44.500 --> 01:20:48.740 align:middle line:90% So it has n minus 1 edges. 01:20:48.740 --> 01:20:51.890 align:middle line:84% And that's what we needed to show. 01:20:51.890 --> 01:20:54.920 align:middle line:84% This is very nice for inducting over trees. 01:20:54.920 --> 01:20:57.470 align:middle line:84% The idea is if you have a tree, you're going to do induction. 01:20:57.470 --> 01:20:59.990 align:middle line:84% You should take a leaf and prune it. 01:20:59.990 --> 01:21:01.877 align:middle line:84% Usually, that's what you want to do. 01:21:01.877 --> 01:21:03.960 align:middle line:84% Sometimes it's useful to know there's two of them. 01:21:03.960 --> 01:21:05.730 align:middle line:84% You could take any vertex and remove it. 01:21:05.730 --> 01:21:07.890 align:middle line:84% But if I took a vertex in the middle of my tree, 01:21:07.890 --> 01:21:09.810 align:middle line:84% it might disconnect into multiple trees. 01:21:09.810 --> 01:21:11.540 align:middle line:84% And then I have to do a lot more work. 01:21:11.540 --> 01:21:13.350 align:middle line:84% By choosing a leaf, when I remove it, 01:21:13.350 --> 01:21:14.910 align:middle line:90% I guarantee I'm still a tree. 01:21:14.910 --> 01:21:16.800 align:middle line:84% And so I can still apply induction. 01:21:16.800 --> 01:21:19.010 align:middle line:90% It's very helpful. 01:21:19.010 --> 01:21:22.670 align:middle line:84% That's lots of fun things about trees. 01:21:22.670 --> 01:21:24.780 align:middle line:84% It turns out the reverse is true. 01:21:24.780 --> 01:21:28.830 align:middle line:84% If you have any connected graph with one fewer edge than vertex, 01:21:28.830 --> 01:21:31.130 align:middle line:90% it must be a tree. 01:21:31.130 --> 01:21:34.800 align:middle line:84% Trees also have unique paths between any two vertices. 01:21:34.800 --> 01:21:37.920 align:middle line:84% And any graph that has unique paths between any two vertices 01:21:37.920 --> 01:21:40.100 align:middle line:90% is a tree. 01:21:40.100 --> 01:21:47.180 align:middle line:84% Other fun things, like they're the minimal connected graphs 01:21:47.180 --> 01:21:49.260 align:middle line:90% or maximal acyclic graphs-- 01:21:49.260 --> 01:21:51.290 align:middle line:84% lots of fun properties, which you can find 01:21:51.290 --> 01:21:53.100 align:middle line:90% in the notes or the textbook. 01:21:53.100 --> 01:21:54.000 align:middle line:90% That's it for trees. 01:21:54.000 --> 01:21:56.740 align:middle line:84% Next time we'll do directed graphs. 01:21:56.740 --> 01:22:02.000 align:middle line:90%