Maximally Connected Graphs and Digraphs

Preface Graphs are often used as a model for networks, for example transport networks, telecommunication systems, a network of servers and so on. Typically, the networks and thus the representing graphs are connected; that means, that there exists a path between all pairs of two vertices in the graph. But sometimes an element of the network fails, for the graphs that means the removal of a vertex or an edge. Clearly, it is desirable that a network stays connected as long as possible in case faults should arise. The graph theoretical parameter edge-connectivity λ(G) equals the minimum number of edges, whose removal disconnects the graph. Analogougsly, the vertex-connectivity κ(G) equals the minimum number of vertices, whose removal disconnects the graph. By removing all vertices or edges adjacent to a vertex of minimum degree δ(G) the resulting graph is always disconnected and hence λ(G) ≤ δ(G) and κ(G) ≤ δ(G). Thus, in order to construct reliable and fault-tolerant networks one is interested in finding sufficient conditions for graphs to satisfy λ(G) = δ(G) and κ(G) = δ(G). For the case that graphs are maximally edge(vertex)-connected, further connectivity parameters are needed in order to investigate the fault-tolerance. Such parameters are for example: A graph is called super-edge-connected, if every minimum edge-cut consists of edges incident to a vertex of minimum degree. The restricted edge-connectivity λ (G) equals the minimum cardinality over all edge-cuts S in a graph G such that there are no isolated vertices in G − S. The p-q-restricted edge(vertex)-connectivity λ p,q (G) (κ p,q) is the minimum cardi-nality of an edge(vertex)-cut S such that one component of G − S contains at least p vertices and another component of G − S contains at least q vertices. The local-edge-connectivity λ(u, v) of two vertices u and v equals the maximum number of edge-disjoint u-v paths. In this thesis, we mainly study sufficient conditions for these connectivity parameters to be maximal. ii After a short introduction to the used notations, we give in Chapter 2, 3, 4, 6 several sufficient conditions for graphs to be maximally edge-connected, super-edge-connected, maximally restricted-edge-connected and maximally local-edge-connected, respectively. Hereby we generalize some known results by Goldsmith and Entringer and by Dankelmann and Volkmann. Furthermore we give analogue results to Xu's theorem for bipartite graphs. In Chapter 5 we characterize the graphs, where the parameter λ p,q exists. The Chapters 7 and 8 deal with …