1. Lecture 5.4: Paths and Connectivity CS 250, Discrete Structures, Fall 2011 Nitesh Saxena *Adopted from previous lectures by Zeph Grunschlag

2. Course Admin -- Homework 5 Due at 11am on Nov 30 (Wed) Covers the chapter on Graphs (lecture 5.*) Thanksgiving is in the way – please try to start early Has a 10-pointer bonus problem too Lecture 5.4 -- Paths and Connectivity Little work will not hurt!

3. Course Admin -- Homework 4 Grading now Expect results in a couple of days Graded HWs will be distributed next Tuesday Solution posted Lecture 5.4 -- Paths and Connectivity

4. Course Admin -- Final Exam Thursday, December 8, 10:45am- 1:15pm, lecture room Heads up! Please mark the date/time/place Emphasis on post mid-term 2 material Coverage: 65% post mid-term 2 (lectures 4.*, 5.*, 6.*), and 35% pre mid-term 2 (lecture 1.*. 2.* and 3.*) Our last lecture will be on December 6 We plan to do a final exam review then Lecture 5.4 -- Paths and Connectivity

5. Outline Paths and Isomorphism Connectivity

6. A visualization of the Linkein Social Graph Lecture 5.4 -- Paths and Connectivity

7. and of the Facebook network Lecture 5.4 -- Paths and Connectivity

8. Recap: Graph Isomorphism DEF: Suppose G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) are pseudographs. Let f :V 1  V 2 be a function s.t.: 1) f is bijective 2) for all vertices u,v in V 1, the number of edges between u and v in G 1 is the same as the number of edges between f (u) and f (v ) in G 2 ; or e(u, v, G 1 ) = e(f(u), f(v), G 2 ) Then f is called an isomorphism and G 1 is said to be isomorphic to G 2. Lecture 5.4 -- Paths and Connectivity

9. Properties of Isomorphisms Two graphs are isomorphic to each other if they satisfy the following properties: same number of vertices same number of edges same degrees at corresponding vertices Any subgraph of one is isomorphic to some subgraph of the other Lecture 5.4 -- Paths and Connectivity

10. Warm-up Exercise Theorem: Isomorphism is an equivalence relation Proof: We need to prove that the isomorphism relation is reflexive, symmetric, and transitive. Let’s use the whiteboard. Lecture 5.4 -- Paths and Connectivity

11. Paths A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges. EG: could get from 1 to 3 circuitously as follows: 1-e 1  2-e 1  1-e 3  3-e 4  2-e 6  2-e 5  2-e 4  3 12 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6 e7e7

12. Paths 1 A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges. EG: could get from 1 to 3 circuitously as follows: 1- e 1  2-e 1  1-e 3  3-e 4  2-e 6  2-e 5  2-e 4  3 2 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6 e7e7

13. Paths A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges. EG: could get from 1 to 3 circuitously as follows: 1-e 1  2- e 1  1-e 3  3-e 4  2-e 6  2-e 5  2-e 4  3 12 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6 e7e7

14. Paths A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges. EG: could get from 1 to 3 circuitously as follows: 1-e 1  2-e 1  1- e 3  3-e 4  2-e 6  2-e 5  2-e 4  3 12 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6 e7e7

15. Paths A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges. EG: could get from 1 to 3 circuitously as follows: 1-e 1  2-e 1  1-e 3  3- e 4  2-e 6  2-e 5  2-e 4  3 12 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6 e7e7

16. Paths A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges. EG: could get from 1 to 3 circuitously as follows: 1-e 1  2-e 1  1-e 3  3-e 4  2- e 6  2-e 5  2-e 4  3 12 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6 e7e7

17. Paths A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges. EG: could get from 1 to 3 circuitously as follows: 1-e 1  2-e 1  1-e 3  3-e 4  2-e 6  2- e 5  2-e 4  3 12 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6 e7e7

18. Paths A path in a graph is a continuous way of getting from one vertex to another by using a sequence of edges. EG: could get from 1 to 3 circuitously as follows: 1-e 1  2-e 1  1-e 3  3-e 4  2-e 6  2-e 5  2- e 4  3 12 34 e1e1 e3e3 e2e2 e4e4 e5e5 e6e6 e7e7

19. Paths in Real (CS) World Linkedin paths (let us see in practice) Facebook paths Internet paths … Lecture 5.4 -- Paths and Connectivity

20. Paths and Isomorphism Paths and circuits can also be serve as a good criteria for determining whether two graphs are isomorphic Look for (simple) circuits of different lengths. Example below: u1u1 u6u6 u2u2 u3u3 u5u5 u4u4 Lecture 5.4 -- Paths and Connectivity

21. Number of Paths of Certain Lengths Adjacency matrix A for a graph depicts all paths of length 1 The matrix A 2 depicts number of paths of length 2 In general, the matrix A k depicts number of paths of length k Lecture 5.4 -- Paths and Connectivity a d c b

22. Number of Paths of Certain Lengths Adjacency matrix A for a graph depicts all paths of length 1 The matrix A 2 depicts number of paths of length 2 In general, the matrix A k depicts number of paths of length k Lecture 5.4 -- Paths and Connectivity a d c b

23. Connectivity DEF: Let G be a pseudograph. Let u and v be vertices. u and v are connected to each other if there is a path in G which starts at u and ends at v. G is said to be connected if all vertices are connected to each other. Note: Any vertex is automatically connected to itself via the empty path. Lecture 5.4 -- Paths and Connectivity

24. Q: Which of the following graphs are connected? Connectivity 12 3 4 Lecture 5.4 -- Paths and Connectivity

25. A: First and second are disconnected. Last is connected. Connectivity 12 3 4 Lecture 5.4 -- Paths and Connectivity

26. A: First and second are disconnected. Last is connected. Connectivity 12 3 4 Lecture 5.4 -- Paths and Connectivity

27. A: First and second are disconnected. Last is connected. Connectivity 12 3 4 Lecture 5.4 -- Paths and Connectivity

28. A: First and second are disconnected. Last is connected. Connectivity 12 3 4 Lecture 5.4 -- Paths and Connectivity

29. A: First and second are disconnected. Last is connected. Connectivity 12 3 4 Lecture 5.4 -- Paths and Connectivity

30. A: First and second are disconnected. Last is connected. Connectivity 12 3 4 Lecture 5.4 -- Paths and Connectivity

31. English Connectivity Puzzle Can define a puzzling graph G as follows: V = {3-letter English words} E : two words are connected if we can get one word from the other by changing a single letter. One small subgraph of G is: Q: Is “fun” connected to “car” ? jobrobjab Lecture 5.4 -- Paths and Connectivity

32. English Connectivity Puzzle A: Yes: fun  fan  far  car Or: fun  fin  bin  ban  bar  car Lecture 5.4 -- Paths and Connectivity

33. Some Little Theorems Thm1: Every connected graph with n vertices has at least n-1 edges Proof: Use proof by mathematical induction Basis Step (n=1): No. of edges is 0, which is less >= 0 (n-1) Induction Step: Assume to be true for n = k and show it to be true for n = k + 1 We assumed that a connected graph with k vertices will have at least k–1 edges. Now, we add a new vertex to this graph to obtain a new graph with k vertices. For the new graph to remain connected, the new vertex should be incident with at least one edge which is also incident with one of the vertices in the old graph. This means that the new graph should have a total of at least (k – 1) + 1 = k edges. This proves the induction step. Finally, combining the basis and induction steps, we get that the theorem is true for all n

34. Some Theorems Thm 2: Vertex connectedness in a simple graph is an equivalence relation Proof: We will show that the “connectedness” relation is reflexive, symmetric and transitive It is reflexive since every vertex is connected to itself via a path of length 0 It is symmetric because if a vertex u is connected to another vertex v, then there exists a path between u and v – just traverse the reverse path It is transitive because if u and v are connected (via path p) and v and w are connected (via path q), then u and w are connected via a path p|q

35. Some Theorems Thm 3: If a connected simple graph G is the union of graphs G1 and G2, then G1 and G2 must have a common vertex Proof: (very simple) let’s use the board. Lecture 5.4 -- Paths and Connectivity

36. Connected Components DEF: A connected component (or just component) in a graph G is a set of vertices such that all vertices in the set are connected to each other and every possible connected vertex is included. Q: What are the connected components of the following graph? 62 4 35 1 7 8 Lecture 5.4 -- Paths and Connectivity

37. Connected Components A: The components are {1,3,5},{2,4,6},{7} and {8} as one can see visually by pulling components apart: 62 4 7 8 35 1 Lecture 5.4 -- Paths and Connectivity

38. Connected Components A: The components are {1,3,5},{2,4,6},{7} and {8} as one can see visually by pulling components apart: 62 4 7 8 35 1 Lecture 5.4 -- Paths and Connectivity

39. Connected Components A: The components are {1,3,5}, {2,4,6}, {7} and {8} as one can see visually by pulling components apart: 62 4 7 8 35 1 Lecture 5.4 -- Paths and Connectivity

40. Degree of Connectivity Not all connected graphs are treated equal! Q: Rate following graphs in terms of their design value for computer networks: 1) 2) 3) 4)

41. Degree of Connectivity A: Want all computers to be connected, even if 1 computer goes down: 1) 2 nd best. However, there is a weak link— “cut vertex” 2) 3 rd best. Connected but any computer can disconnect 3) Worst! Already disconnected 4) Best! Network dies only with 2 bad computers Lecture 5.4 -- Paths and Connectivity

42. Degree of Connectivity The network is best because it can only become disconnected when 2 vertices are removed. In other words, it is 2-connected. Formally: DEF: A connected simple graph with 3 or more vertices is 2-connected if it remains connected when any vertex is removed. When the graph is not 2-connected, we call the disconnecting vertex a cut vertex. Lecture 5.4 -- Paths and Connectivity

43. Degree of Connectivity There is also a notion of N-Connectivity where we require at least N vertices to be removed to disconnect the graph. Lecture 5.4 -- Paths and Connectivity

44. Connectivity in Directed Graphs In directed graphs may be able to find a path from a to b but not from b to a. However, Connectivity was a symmetric concept for undirected graphs. So how to define directed Connectivity is non- obvious: 1) Should we ignore directions? 2) Should we insist that we can get from a to b in actual digraph? 3) Should we insist that we can get from a to b and that we can get from b to a? Lecture 5.4 -- Paths and Connectivity

45. Connectivity in Directed Graphs Resolution: Don’t bother choosing which definition is better. Just define to separate concepts: 1) Weakly connected : can get from a to b in underlying undirected graph 2) Semi-connected (my terminology): can get from a to b OR from b to a in digraph 3) Strongly connected : can get from a to b AND from b to a in the digraph DEF: A graph is strongly (resp. semi, resp. weakly) connected if every pair of vertices is connected in the same sense. Lecture 5.4 -- Paths and Connectivity

46. Connectivity in Directed Graphs Q: Classify the connectivity of each graph. Lecture 5.4 -- Paths and Connectivity

47. Connectivity in Directed Graphs A: semi weakstrong Lecture 5.4 -- Paths and Connectivity

48. Today’s Reading Rosen 10.3 and 10.4 Little work will not hurt!