Generalized edge connectivity in graphs

Abstract

It is well known that graph theory plays a key role in the analysis and design of reliable networks. For example, given a positive integer n, what is the minimum number of edges (links) to be removed from a graph so that the resulting graph has at least n components? This problem is solved here. The edge connectivity of a connected graph is the minimum number of edges whose deletion produces a graph with two components. For an integer n, the n-edge-connectivity of a connected graph is the smallest number of edges whose removal results in a graph with n components. This is called the generalized edge connectivity. Several properties of generalized edge connectivity are derived, and the problem of finding graphs which are optimal with respect to generalized edge connectivity is discussed. We also present an optimal model of an interconnection network with a given generalized edge connectivity such that the edge connectivity of the network is maximized while number of links is minimized. For a positive integer k, a graph is (k,k)-edge-connected if the k-edge-connectivity of the graphs is equal to k. For graphs without bridges, we present a structural characterization of minimally (k,k)-edge-connected graph. Further, we extend the former characterization of minimally (2,2)-edge-connected graph in [J. of Graph theory, 3 (1979) 15-22].