Nonexistence of uniformly most reliable two-terminal graphs

Abstract

A two-terminal graph G=(V,E) is a simple and undirected graph with two specified target vertices s and t in V. In G, if each edge survives independently with a fixed probability p, the two-terminal reliability is the probability that two target vertices are connected. A two-terminal graph is uniformly most reliable if its reliability is not less than the reliability of any other graph with same number of vertices and edges for all p. Betrand et al. proved that there is no uniformly most reliable two-terminal graph if either n≥11 and 20≤m≤3n−9 or n≥8 and (n2)−⌊(n−2)/2⌋≤m≤(n2)−2. In this paper, we further prove that there is no uniformly most reliable two-terminal graph if n≥6 and 3n−6<m≤(n2)−2 in a different way.