Domination Invariant of a Diameter Constrained Network Reliability Model

Abstract

Let G=(V,E) be a digraph with a distinguished set of terminal vertices K in V and a vertex s in K. We define 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, R_\{s,K\}(G,D) is defined as the probability that surviving arcs span a subgraph whose s,K-diameter does not exceed D. The Diameter-constrained network reliability is a special case of coherent system models, where the domination invariant has played an important role, both theoretically and for developing algorithms for reliability computation. In this work, we completely characterize the domination of diameter-constrained network models, giving a simple rule for computing its value: if the digraph either has an irrelevant edge, includes a dicycle or includes a dipath from $s$ to a node in K longer than D, its domination is 0; otherwise, its domination is -1 to the power |E|-|V|+1.