Diameter-related properties of graphs and applications to network reliability theory

Given an undirected graph G = (V,E), two distinguished vertices s and t of G, and a diameter bound D, a D-s, t-path is a path between s and t composed of at most D edges. An edge e is called D-irrelevant if does not belong to any D-s, t-path of G. In this paper we study the problem of efficiently detecting D-irrelevant edges and also study the computational complexity of diameter-related problems in graphs. Detection and subsequent deletion ofD-irrelevant edges have been shown to be fundamental in reducing the computational effort to evaluate the Source-to-terminal Diameter-Constrained reliability of a graph G, R{s,t}(G,D), which is defined as the probability that at least a path between s and t, with at most D edges, survives after deletion of the failed edges (under the assumption that edges fail independently and nodes are perfectly reliable). Among other results, we present sufficient conditions to efficiently recognize irrelevant edges and we present computational results illustrating the importance of embedding a procedure to detect irrelevant edges based on these conditions, within the frame of an algorithm to calculate R{s,t}(G,D), built on a theorem of Moskowitz. These results yield a research path for the theoretical study of the problem of determining families of topologies in which R{s,t}(G,D) can be computed in polynomial time, as the general problem of evaluating this reliability measure is NP-Hard. Key–Words: Network reliability, diameter constraint, paths, factoring, topological reductions.