Uniformly Least Reliable Graphs in Class \(\Omega(n,e)\) as \(e \leq n + 1\)

Abstract

We consider the undirected simple connected graph for which edges fail independently of each other with equal probability \(1 – p\) and nodes are perfect. The all-terminal reliability of a graph \(G\) is the probability that the spanning subgraph of surviving edges is connected, denoted as \(R(G,p)\). Graph \(G \in \Omega(n,e)\) is said to be uniformly least reliable if \(R(G,p) \leq R(G’,p)\) for all \(G’ \in \Omega(n,e)\), and for all edge failure probabilities \(0 < 1 – p < 1\). In this paper, we prove the existence of uniformly least reliable graphs in the class \(\Omega(n,e)\) for \(e \leq n + 1\) and give their topologies.