ON THE EXISTENCE OF UNIFORMLY OPTIMALLY RELIABLE NETWORKS

A well-known model for network reliability studies consists of an undirected graph with perfectly reliable nodes and equal and independent edge failure probabilities. The measure of reliability is then defined to be the probability that the graph is connected. A well-defined synthesis problem is to find the graph that minimizes the failure probability given the number of nodes n, the number of edges e, and the edge failure rate rho. In this work, we consider the possibility of the existence of a fixed graph that is optimal for all possible rho. It is simple to verify that such graphs exist when e = n - 1, or n. Herein, we show that they also exist when e = n + 1 and n + 2.