Connectivity and Paths 報告人:林清池
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.
Example
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
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.
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.
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.
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.
Harary graphs If k and n are both odd, index the vertices by the integers modulo n. Construct form by adding the edges for
Harary graphs Theorem., and hence the minimum number of edges is a k- connected graph on n vertices is
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.
Harary graphs u v
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.
Edge-Connectivity An edge cut is an edge set of the form where is a nonempty proper subset of and denotes Disconnecting setEdge cut
Theorem If G is a simple graph, then Proof:, trivial. Case 1: if every vertex of is adjacent to every vertex of, then
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.
Example
Theorem If G is a 3-regular graph, then Proof:
Theorem If G is a 3-regular graph, then Proof:
Theorem If G is a 3-regular graph, then Proof:
Theorem If G is a 3-regular graph, then Proof:
Definition A Bond is a minimal nonempty edge cut. Here “ minimal ” means that no proper nonempty subset is also an edge cut.
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.
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.
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.
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.
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.
Algorithm Computing the blocks of a graph.
Algorithm Computing the blocks of a graph.
Algorithm Computing the blocks of a graph.
Algorithm Computing the blocks of a graph.
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 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 Basis step. The graph G-uv is connected. Induction step. Let w be the vertex before v on a shortest u,v path;
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.
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
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.
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.
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 …
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.
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.
Theorem
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.
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, …
Theorem Proof:
Theorem Proof:
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.
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
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, 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
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
Example Although, it takes four edges to break all w, z-paths, and there are four pairwise edge-disjoint w, z-paths.