More reliable graphs are not always stronger

Abstract

AbstractA graph is stronger than a graph if has at least as many connected spanning subgraphs of size as for any positive integer . Counting the number of connected spanning subgraphs of fixed size allows us to compute the reliability of a graph. Formally, the reliability polynomial of a graph is the probability that the graph is connected when each edge is deleted independently with the same fixed probability. A graph is uniformly more reliable than if its reliability polynomial is greater than or equal to the reliability polynomial of for all probabilities. As a direct consequence of the definition, a sufficient condition for to be uniformly more reliable than is for to be stronger than . In this paper, we show that the sufficient condition is not necessary by providing an example of two infinite families of graphs, and , such that is uniformly more reliable than but is not stronger than .