Efflcient evaluation of a Diameter-Constrained reliability measure of some families of graphs

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 or vertex u are called Dirrelevant if e or u does not belong to anyD-s, t-path ofG. In this paper we study the problem of efficiently detecting D-irrelevant edges and vertices and also study the computational complexity of diameter-related problems in graphs. Detection and subsequent deletion of D-irrelevant edges and vertices 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). Even though the computation of R{s,t}(G,D) has be shown to be NP-Hard for fixed values of the diameter bound D (i.e., D is independent of the size of a given graph), we show that the calculation of this reliability measure can be determined in time 2 ) for some families of graphs, yielding an efficient determination of the reliability for small values of D.