1. Connectivity and Paths 報告人：林清池

2. Connectivity A separating set of a graph G is a set such that G-S has more than one component. The connectivity of G, is the minimum size of a vertex set S such that G-S is disconnected or has only one vertex. A graph G is k-connected if its connectivity is at least k.

3. Example

4. Hypercube The K-dimensional cube is the simple graph whose vertices are the k-tuples with entries in and whose edges are the pairs of k-tuples that differ in exactly one position. 111011 101001 110010 100000

5. The neighbors of one vertex in form a separating set, so. To prove, we show that every separating set has size at least. Prove by induction on. Basis step: For, is a complete graph with vertices and has connectivity.

6. An example: Induction step: Let S be a vertex cut in Case 1: If Q-S is connected and Q ’ -S is connected, then, for. Case 2: If Q-S is disconnected, which means S has at least k-1 vertex in Q. And, S must also contain a vertex of. We have.

7. Harary graphs Given k <n, place n vertices around a circle. If k is even, form by making each vertex adjacent to the nearest k/2 vertices in each direction around the circle.

8. Harary graphs If k is odd and n is even, form by making each vertex adjacent to the nearest (k-1)/2 vertices in each direction around the circle and to the diametrically opposite vertex.

9. Harary graphs If k and n are both odd, index the vertices by the integers modulo n. Construct form by adding the edges for 2 1 3 4 5 6 7 08

10. Harary graphs Theorem., and hence the minimum number of edges is a k- connected graph on n vertices is

11. Harary graphs Proof. (Only the even case k =2r. Pigeonhole) Since, it suffices to prove Clockwise u,v paths and counterclockwise u,v paths. Let A and B be the sets of internal vertices on these two paths. One of {A, B} has fewer that k/2 vertices. Thus, we can find a u,v path in G-S via the set A or B in which S has fewer than k/2 vertices.

12. Harary graphs u v

13. Edge-Connectivity A disconnecting set of a graph G is a set such that G-F has more than one component. The edge-connectivity of G, is the minimum size of a disconnecting set. A graph G is k-edge-connected if every disconnecting set has at least k edges.

14. Edge-Connectivity An edge cut is an edge set of the form where is a nonempty proper subset of and denotes Disconnecting setEdge cut

15. Theorem If G is a simple graph, then Proof:, trivial. Case 1: if every vertex of is adjacent to every vertex of, then

16. Theorem Case 2:, with : consist of all neighbors of in and all vertices of with neighbors in.  is a separating set picking the red edges yields |T| distinct edges. 

17. Example

18. Theorem If G is a 3-regular graph, then Proof:

19. Theorem If G is a 3-regular graph, then Proof:

20. Theorem If G is a 3-regular graph, then Proof:

21. Theorem If G is a 3-regular graph, then Proof:

22. Definition A Bond is a minimal nonempty edge cut. Here “ minimal ” means that no proper nonempty subset is also an edge cut.

23. Proposition If is a connected graph, then an edge cut is a bound if and only if has exactly two components. Proof:  is a subset of. is connected.

24. Proposition If is a connected graph, then an edge cut is a bound if and only if has exactly two components. Proof:  Suppose has more than two component. and are proper subsets of, so is not a bound.

25. Definition A Block of a graph is a maximal connected subgraph of G that has no cut-vertex. A connected graph with on cut-vertex need not be 2-connected, since it can be or.

26. Proposition Two blocks in a graph share at most one vertex. Proof: Suppose for a contradiction. and have at least two common vertices. Since the blocks have at least two common vertices, deleting one singe vertex, what remains is connected. A contradiction.

27. Definition The block-cutpoint graph of a graph G is a bipartite graph H in which one partite set consists of the cut-vertices of G, and the other has a vertex for each block of G. We include as an edge of H if and only if.

28. Algorithm Computing the blocks of a graph.

29. Algorithm Computing the blocks of a graph.

30. Algorithm Computing the blocks of a graph.

31. Algorithm Computing the blocks of a graph.

32. Definition Two paths from u to v are internally disjoint if they have no common internal vertex.

33. Theorem G is 2-connected if and only if for each for each pair there exist internally disjoint u,v paths in G. Proof:  Since for every pair u,v, G has internally disjoint u,v paths, deletion of one vertex cannot make any vertex unreachable from any other.

34. Theorem  Prove by induction on Basis step. The graph G-uv is connected. Induction step. Let w be the vertex before v on a shortest u,v path;

35. Theorem Case 1: if, done. Case 2: G-w is connected and contains a u,v path R. If R avoids P or Q, done. Case 3: Let z be the last vertex of R.

36. Expansion Lemma If G is a k-connected, and G ’ is obtained from G by adding a new vertex y with at least k neighbors in G, then G ’ is k-connected. Case 1: if, then Case 2: if and, then Case 3: and lie in a single component of, then

37. Theorem For a graph G with at least three vertices, the following condition are equivalent. G is connected and no cut-vertex. For, there are internally disjoint x, y paths. For, there is a cycle through x and y., and every pair of edges in G lies on a common cycle.

38. Definition In a graph G, subdivision of an edge uv is the operation of replacing uv with a path u, w, v through a new vertex w.

39. Corollary If G is a 2-connected, then the graph G ’ obtained by subdividing an edge of G is 2-connected. Proof: It suffices to find a cycle through arbitrary edges e,f of G ’. Since G is 2-connected, any two edges of G lie on a common cycle. Case 1: if a cycle through them in G uses uv, then replace the edge uv with a path u,w,v. Case 2: if and, then … Case 3: if, then …

40. Definition An ear of a graph G is a maximal path whose internal vertices have degree 2 in G. An ear decomposition of G is a decomposition such that is a cycle and for is an ear of.

41. Theorem A graph is 2-connected if and only if it has an ear decomposition. Proof:  Since cycles are 2-connected, it suffices to show that adding an ear preserves 2-connectedness. Trivial. 

42. Theorem

43. Definition An close ear in a graph G is a cycle C such that all vertices of C expect one have degree 2 in G An close-ear decomposition of G is a decomposition such that is a cycle and for is either an (open) ear or a closed ear in.

44. Theorem A graph is 2-edge-connected if and only if it has an closed-ear decomposition. Proof:  G is 2-edge-connected if and only if every edge lies on a cycle. Case 1: when adding a closed ear, Trivial. Case 2: when adding a open ear, …

45. Theorem Proof: 

46. Theorem Proof: 

47. Connectivity of Digraphs A separating set of a digraph D is a set such that D-S is not strongly connected. The connectivity of G, is the minimum size of a vertex set S such that D-S is not strong or has only one vertex. A graph G is k-connected if its connectivity is at least k.

48. Edge-Connectivity of Digraphs For vertex sets S, T in a digraph D, let [S,T] denote the set of edges with tail in S and head in T. An edge cut is an edge set of the form for some. A diagraph is k-edge-connected if every edge cut has at least k edges. The minimum size of an edge cut is the edge- connected

49. Proposition Adding a directed ear to a strong digraph produces a larger strong digraph.

50. Theorem A graph has a strong orientation if and only if it is 2-edge-connected. Proof:  If G has a cut-edge xy oriented from x to y in an orientation D, then y cannot reach x in D.  1.) G has a closed-ear decomposition. 2.) Orient the initial cycle consistently to obtain a strong diagraph. 3.) Directing new ear consistently.

51. Definition Given, a set is an x,y separator or x, y-cut if G-S has no x, y-path. Let be the minimum size of an x,y-cut. Let be the maximum size of a set of pairwise internally disjoint x, y-paths. For, an X, Y-path is a graph having first vertex in X, last vertex in Y, and no other vertex in

52. Remark An x, y-cut must contain an internal vertex of every x, y-path, and no vertex can cut two internally disjoint x,y-paths. Therefore, always

53. Example Although, it takes four edges to break all w, z-paths, and there are four pairwise edge-disjoint w, z-paths.