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This chapter explores the concept of connectivity in graphs, examining how the deletion of edges and vertices affects the structure. It classifies graphs based on their connectivity: G1 is vulnerable to edge deletion, while G2 is more resilient but can still be disconnected by removing its cut vertex. Fundamental theorems outline the conditions for cut edges and vertices, leading to discussions on vertex and edge connectivity. The chapter concludes with an application of these concepts to reliable communication networks, emphasizing the importance of connectivity in system robustness.
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9.1 Connectivity Consider the following graphs: • G1: Deleting any edge makes it disconnected. • G2: Cannot be disconnected by deletion of any edge; can be disconnected by deleting its cut vertex; • Intuitively, G2 is more connected than G1, G3 is more connected thant G2, and G4 is the most connected one.
9.1 Cut edges and cut vertices A cut edge of G is an edge such that G-e has more components that G. Theorem 9.1 Let G be a connected graph. The following are equivalent: • An edge e of G is a cut edge • e is not contained in any cycle of G. • There are two vertices u and w such that e is on every path connecting u and w.
9.1 Cut edges and cut vertices Let G be a nontrivial and loopless graph. A vertex v of G is a cut vertex if G-v has more components than G. Theorem 9.2 Let G be a connected graph. The following propositions are equivalent: 1. A vertex v is a cut vertex of G 2. There are two distinct vertices u and w such that every path between u and w passes v; 3. The vertices of G can be partitioned into two disjoint vertex sets U and W such that every path between uU and wW passes v.
9.1 Vertex cut and connectivity A vertex cut of G is a subset V’ of V such that G-V’ is disconnected. The connectivity , (G), is the smallest number of vertices in any vertex cut of G. • A complete graph has no vertex cut. Define (Kn)=n-1; • For disconnected graph G, define (G) = 0; • G is said to be k-connected if (G)k; • It is easy to see that all nontrivial connected graphs are 1-connected. • (G)=1 if and only if G=K2 or G has a cut vertex.
9.1 Edge cut and edge connectivity Let [S,S’] denote the set of edges with one end in S and the other end in S’. Let G be graph on n2 vertices. An edge cut is a subset E’ of E(G) of the form [S,S’], where S is a nonempty proper set of V and S’=V-S. If G is nontrivial and E’ is an edge cut of G, then G-E’ is disconnected. The edge connectivity, (G), is the smallest number of edges in any edge cut.
9.1 Edge cut and edge connectivity • For trivial and disconnected graph G, define (G)=0; • G is said to be k-edge-connected if (G)k ; • All nontrivial connected graphs are 1-edge-connected. Theorem 9.3 For any connected graph G (G) (G)(G) where (G) is the smallest vertex degree of G.
9.2 Menger’s theorem Theorem A graph G is k-edge-connected if and only if any two distinct vertices of G are connected by at least k edge-disjoint paths. Proof：If there are two vertices which are connected by less than k edge-disjoint paths, then G is not k-edge-connected. On the other hand, if G is not k-edge-connected, there are edge cut that contains less than k edges, hence there are two vertices which are connected by less than k edge-disjoint paths. Theorem A graph with nk+1 is k-connected if and only if any two distinct vertices of G are connected by at least k vertex-disjoint paths.
9.3 Reliable communication networks • A graph representing a communication network, the connectivity (or edge-connectivity) becomes the smallest number of stations (or links) whose breakdown would jeopardise the system. • The higher the connectivity and edge connectivity, the more reliable the network. • Let k be a given positive integer and let G be a weighted graph. Determine a minimum-weight k-connected spanning subgraph of G. • For k=1, this is solved by Kruskal’s algorithm, for example. For k>1, the problem is unsolved.
