On the characterization of the source-to-all-terminal diameter-constrained reliability domination

Abstract

Let G = (V;E) be a digraph with a distinguished set of terminal vertices K V and a vertex s 2 K . We de ne the s;K-diameter of G as the maximum distance between s and any of vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the Diameter-constrained s;K-terminal reliability of G, Rs;K(G;D), is de ned as the probability that surviving arcs span a subgraph whose s;K- diameter does not exceed D [5, 11]. A graph invariant called the domination of a graph G was introduced by Satyanarayana and Prabhakar [13] to generate the non-canceling terms of the classical reliability expres- sion, Rs;K(G), based on the same reliability model (i.e. arcs fail randomly and indepen- dently and where nodes are perfect), and de ned as the probability that the surviving arcs span a subgraph of G with unconstrained nite s;K-diameter. This result allowed the generation of rapid algorithms for the computation of Rs;K(G). In this paper we present a characterization of the diameter-constrained s;K-terminal reliability domination of a digraph G = (V;E) with terminal set K = V , and for any diameter bound D, and, as a result, we solve the classical reliability domination, as a speci c case. Moreover we also present a rapid algorithm for the evaluation of Rs;V (G;D).