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Connectivity and Paths 報告人:林清池 Connectivity A separating set of a graph G is a set S V (G ) such that G-S has more than one component. The connectivity of G, (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. Example ( Kn ) n 1. ( K m ,n ) min{ m, n}. ( K1 ) 0. Hypercube The K-dimensional cube Qk is the simple graph whose vertices are the k-tuples with entries in {0,1} and whose edges are the pairs of k-tuples that differ in exactly one position. 011 010 Q3 110 001 000 111 100 101 (Qk ) k The neighbors of one vertex in Qk form a separating set, so (Qk ). Tok prove , we (Qk ) that k show every separating set has size at k least . Prove by induction on . Basis step: For k 1 , Qk is a complete graph with k 1 vertices and has connectivity k . An example: (Qk ) k Induction step: Let S be a vertex cut in Qk Case 1: If Q-S is connected and Q’-S is k 1 connected, then | S | 2 k , for k 2 . 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 Q ' . We have | S | k . Q Q' Q Q' Qk 1 Harary graphs H k ,n Given k <n, place n vertices around a circle. If k is even, form H k ,n by making each vertex adjacent to the nearest k/2 vertices in each direction around the circle. H 4 ,8 Harary graphs H k ,n If k is odd and n is even, form H k ,n by making each vertex adjacent to the nearest (k-1)/2 vertices in each direction around the circle and to the diametrically opposite vertex. H 5,8 Harary graphs H k ,n If k and n are both odd, index the vertices by the integers modulo n. Construct H k ,n form H k 1,n by adding the edges i i (n 1) / 2 for 0 i (n 1) / 2. 4 5 3 2 6 1 7 0 8 H 5, 9 Harary graphs Theorem. ( H k ,n ) k , and hence the minimum number of edges is a kconnected graph on n vertices is kn / 2 Harary graphs Proof. (Only the even case k =2r. Pigeonhole) Since (G ) k , it suffices to prove (G ) k . 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. Harary graphs H k ,n u H 4 ,8 v Edge-Connectivity A disconnecting set of a graph G is a set F E (G )such that G-F has more than one component. The edge-connectivity of G, ' (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. Edge-Connectivity An edge cut is an edge set of the form S, S where S is a nonempty proper subset of V (G ) and S denotes V (G ) S S S Disconnecting set Edge cut Theorem If G is a simple graph, then (G ) ' (G ) (G ) Proof: ' (G ) (G ) , trivial. Case 1: if every vertex of S is adjacent to every vertex of S , then | S , S || S || S | n(G ) 1 (G ) S S Theorem Case 2: x S , y S with xy E (G ) T : consist of all neighbors of x in S and all vertices of S {x} T with neighbors in S . is a separating set picking the red edges yields |T| distinct edges. ' (G ) S , S | T | (G ) x S T T T T T y S Example (G ) 1, ' (G ) 2, (G ) 3 Theorem If G is a 3-regular graph, then (G ) ' (G ) Proof: S H H' Theorem If G is a 3-regular graph, then (G ) ' (G ) Proof: S H H' Theorem If G is a 3-regular graph, then (G ) ' (G ) Proof: S H H' Theorem If G is a 3-regular graph, then (G ) ' (G ) Proof: S H H' Definition A Bond is a minimal nonempty edge cut. Here “minimal” means that no proper nonempty subset is also an edge cut. Proposition If G is a connected graph, then an edge cut F is a bound if and only if G F has exactly two components. Proof: F ' is a subset of F . G F ' is connected. G F' F S, S S S Proposition If G is a connected graph, then an edge cut F is a bound if and only if G F has exactly two components. Proof: Suppose G F has more than two component. B, B and A, A are proper subsets of F , so F is not a bound. A S B S 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 K1 or K2 . Proposition Two blocks in a graph share at most one vertex. Proof: Suppose for a contradiction. B1 and B2 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. 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 bi for each block Bi of G. We include vbi as an edge of H if and only if v Bi . b c h i a d b5 e x g b1 f j b3 b2 a e x b4 Algorithm Computing the blocks of a graph. x b c a d a h i e x g f b j i e c f j g d h Algorithm Computing the blocks of a graph. x b c a h i a e j g i e d x f f j g h Algorithm Computing the blocks of a graph. x b c a h i a e j g e d x f j i Algorithm Computing the blocks of a graph. x b c a h i a e j g e d x f j Definition Two paths from u to v are internally disjoint if they have no common internal vertex. Theorem G is 2-connected if and only if for each for each pair u, v V (G ) 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. Theorem Prove by induction on d (u, v ) Basis step. The graph G-uv is connected. Induction step. Let w be the vertex before v on a shortest u,v path; P u w Q v Theorem Case 1: if u, v V ( P) V (Q ) , 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. P u Q R P z w v u Q R z w v 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 y S, then | S | k 1. Case 2: if y S and N ( y ) S , then | S | k . Case 3: y and N ( y ) S lie in a single component of G'S , then | S | k . G y Theorem For a graph G with at least three vertices, the following condition are equivalent. G is connected and no cut-vertex. For x, y V (G ), there are internally disjoint x, y paths. For x, y V (G ), there is a cycle through x and y. (G ) 1 , and every pair of edges in G lies on a common cycle. 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. u v u w v 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 e E (G ) and f {uw, wv}, then … Case 3: if {e, f } {uw, wv}, then … 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 P0 ,, Pk such that P0 is a cycle and Pi for i 1 is an ear of P0 Pk . P 3 P0 P2 P4 P1 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. P3 P0 P2 P4 P1 Theorem P3 P0 P2 P1 P3 P4 P0 P2 P4 P1 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 decomposition P0 ,, Pk such that P0 and G is a is a cycle Pi for i 1 is either an (open) ear or a closed ear in P0 Pk . P3 P0 P5 P2 P4 P1 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. P3 Case 2: when adding a open ear P2 , … P0 P2 P4 P1 Theorem Proof: P3 P0 P5 P2 P4 P3 P1 P0 P5 P2 P4 P1 Theorem Proof: P3 P0 P2 P4 P1 Connectivity of Digraphs A separating set of a digraph D is a set S V (D ) such that D-S is not strongly connected. The connectivity of G, (D ) 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. 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 S, S for some S V (D ) . A diagraph is k-edge-connected if every edge cut has at least k edges. The minimum size of an edge cut is the edgeconnected ' ( D ). Proposition Adding a directed ear to a strong digraph produces a larger strong digraph. 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. Definition Given x, y V (G ) , a set S V (G ) {x, y} is an x,y separator or x, y-cut if G-S has no x, y-path. Let ( x, y ) be the minimum size of an x,y-cut. Let ( x , y ) be the maximum size of a set of pairwise internally disjoint x, y-paths. For X , Y V ,(Gan ) X, Y-path is a graph having first vertex in X, last vertex in Y, and no other vertex in X Y. 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 ( x, y ) ( x, y ). Example Although ( w, z ) 3 , it takes four edges to break all w, z-paths, and there are four pairwise edge-disjoint w, z-paths. x w z y ( x, y ) ( x, y ) 4 ( w, z ) ( w, z ) 3