What is a graph ?

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

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

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 5 4

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 5 4

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

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

7. Representing a graph adjacency matrix 3 2 1 5 4 space = (V2)

8. Representing a graph adjacency matrix 3 2 1 5 4 space = (V2) is {u,v} an edge ? (?) list neighbors of v (?)

9. Representing a graph adjacency matrix 3 2 1 5 4 space = (V2) 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 3 2 1 5 4 1: 3,5 2: 3,4 3: 1,2,4 4: 2,3 5: 1 space = (E)

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

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

14. Representing a graph adjacency lists space = (E) 1: 3,5 2: 3,4 3: 1,2,4 4: 2,3 5: 1 is {u,v} in E ? (min{du,dv}) neigbors of v ? (dv) adjacency matrix space = (V2) 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) G – undirected graph, V={1,...,n} seen[v] = false for all v  V Q=queue (FIFO) 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)

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

19. DFS G – undirected graph, V={1,...,n} visited[v] = false for all v  V 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++

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  Iv := [pre[v],post[v]] “interval property” for u,v V either * Iv and Iu 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 D C

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 tree edges B D C

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

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

25. Digraphs (directed graphs) a path = sequence of vertices v1,v2,...,vk such that (v1,v2) E, ... , (vk-1,vk) E

26. DAGs (acyclic digraphs) a cycle = sequence of vertices v1,v2,...,vk such that (v1,v2) E, ... , (vk-1,vk),(vk,v1) 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 G = digraph, V={1,...,n} visited[v] = false for all v  V 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++

29. DFS on digraphs A B D C

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

31. DFS on digraphs A tree edge B D C

32. DFS on digraphs A tree edge B D C

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

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

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

36. Relationships between the intervals? back edge A tree edge B forward edge D C cross 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. Lemma: in a DAG every edge goes to a vertex with lower post 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++

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[])