1. What is a graph ? 1 2 3 4 5

2. G=(V,E) V = a set of vertices E = a set of edges edge = unordered pair of vertices 1 2 3 4 5

3. What is a graph ? G=(V,E) V = {1,2,3,4,5} E = {{1,5}, {3,5}, {2,3}, {2,4}, {3,4}} 1 2 3 4 5

4. What is a graph ? G=(V,E) V = {1,2,3,4,5} E = {{1,5}, {2,3}, {2,4}, {3,4}} 1 2 3 4 5

5. Connectedness 1 2 3 4 5 1 2 3 4 5 connected not connected How can we check if a graph is connected?

6. Representing a graph |V| * |V| symmetric matrix A with A i,j = 1 if {i,j} E A i,j = 0 otherwise adjacency matrix

7. Representing a graph adjacency matrix 1 2 3 4 5 00101 00110 11010 01100 10000 space = (V 2 )

8. Representing a graph adjacency matrix 1 2 3 4 5 00101 00110 11010 01100 10000 space = (V 2 ) is {u,v} an edge ? (?) list neighbors of v (?)

9. Representing a graph adjacency matrix 1 2 3 4 5 00101 00110 11010 01100 10000 space = (V 2 ) is {u,v} an edge ? (1) list neighbors of v (n)

10. Representing a graph adjacency lists for each vertex v V linked list of neighbors of v

11. Representing a graph adjacency lists 1 2 3 4 5 space = (E) 1: 3,5 2: 3,4 3: 1,2,4 4: 2,3 5: 1

12. Representing a graph adjacency lists 1 2 3 4 5 space = (E) 1: 3,5 2: 3,4 3: 1,2,4 4: 2,3 5: 1 is {u,v} an edge ? (?) list neighbors of v (?)

13. Representing a graph adjacency lists 1 2 3 4 5 space = (E) 1: 3,5 2: 3,4 3: 1,2,4 4: 2,3 5: 1 is {u,v} an edge ? (min(d v,d u )) list neighbors of v (d v )

14. Representing a graph adjacency lists space = (E) 1: 3,5 2: 3,4 3: 1,2,4 4: 2,3 5: 1 00101 00110 11010 01100 10000 space = (V 2 ) adjacency matrix is {u,v} in E ? (min{d u,d v }) neigbors of v ? (d v ) is {u,v} in E ? (1) neigbors of v ? (n)

15. Counting connected components How can we check if a graph is connected? INPUT: graph G given by adjacency list OUTPUT: number of components of G

16. BFS (G,v) seen[v] true enqueue(Q,v) while Q not empty do w dequeue(Q) for each neighbor u of w if not seen[u] then seen[u] true enqueue(Q,u) G – undirected graph, V={1,...,n} seen[v] = false for all v V Q=queue (FIFO)

17. Counting connected components C 0 for all v V do seen[v] false for all v V do if not seen[v] then C++ BFS(G,v) output G has C connected components

18. DFS explore(G,v) visited[v] true for each neighbor u of v if not visited(u) then explore(G,u) G – undirected graph, V={1,...,n} visited[v] = false for all v V

19. DFS explore(G,v) visited[v] true pre[v] clock; clock++ for each neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++ G – undirected graph, V={1,...,n} visited[v] = false for all v V

20. DFS explore(G,v) visited[v] true pre[v] clock; clock++ for each neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++ vertex I v := [pre[v],post[v]] interval property for u,v V either * I v and I u are disjoint, or * one is contained in the other

21. DFS explore(G,v) visited[v] true pre[v] clock; clock++ for each neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++ A B C D

22. DFS explore(G,v) visited[v] true pre[v] clock; clock++ for each neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++ A B C D tree edges

23. Digraphs (directed graphs) G=(V,E) V = a set of vertices E = a set of edges edge = ordered pair of vertices u v (u,v)

24. Digraphs (directed graphs) adjacency lists for each vertex v V linked list of out-neighbors of v |V| * |V| matrix A with A i,j = 1 if (i,j) E A i,j = 0 otherwise adjacency matrix

25. a path = sequence of vertices v 1,v 2,...,v k such that (v 1,v 2 ) E,..., (v k-1,v k ) E Digraphs (directed graphs)

26. DAGs (acyclic digraphs) a cycle = sequence of vertices v 1,v 2,...,v k such that (v 1,v 2 ) E,..., (v k-1,v k ),(v k,v 1 ) E DAG = digraph with no cycle

27. Topological sort (linearization) INPUT: DAG G given by adjacency list OUTPUT: ordering of vertices such that edges go forward

28. DFS on digraphs explore(G,v) visited[v] true pre[v] clock; clock++ for each out-neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++ G = digraph, V={1,...,n} visited[v] = false for all v V

29. DFS on digraphs A B C D

30. A B C D root descendant, ancestor child, parent

31. DFS on digraphs A B C D tree edge

32. DFS on digraphs A B C D tree edge

33. DFS on digraphs A B C D tree edge back edge

34. DFS on digraphs A B C D tree edge back edge cross edge

35. DFS on digraphs A B C D tree edge back edge cross edge forward edge

36. Relationships between the intervals? A B C D tree edge back edge cross edge forward edge

37. Topological sort using DFS Lemma: digraph is a DAG if and only if DFS has a back edge.

38. Topological sort using DFS Lemma: digraph is a DAG if and only if DFS has a back edge. explore(G,v) visited[v] true pre[v] clock; clock++ for each neighbor u of v if not visited(u) then explore(G,u) post[v] clock; clock++ Lemma: in a DAG every edge goes to a vertex with lower post

39. (strong) connectedness a digraph G is strongly connected if for every u,v V there exists a path from u to v in G

40. (strong) connectedness How to check if a digraph is strongly connected?

41. (strong) connectedness How to check if a digraph is strongly connected? for every u V do DFS(G,u) check if every v V was visited

42. (strong) connectedness How to check if a digraph is strongly connected? pick some u V DFS(G,u) check if every v V was visited DFS(reverse(G),u) check if every v V was visited

43. Strongly connected components DAG of strongly connected components

44. Strongly connected components Lemma: G and reverse(G) have the same strongly connected components.

45. Strongly connected components DAG of strongly connected components

46. Strongly connected components for all v V do color[v] white for all v V do if color[v]=white then DFS(reverse(G),v) DFS(G,u) (vertices in order post[])