On the split reliability of graphs

A common model of robustness of a graph against random failures has all vertices operational, but the edges independently operational with probability p. One can ask for the probability that all vertices can communicate (all-terminal reliability) or that two specific vertices (or terminals) can communicate with each other (two-terminal reliability). A relatively new measure is split reliability, where for two fixed vertices s and t, we consider the probability that every vertex communicates with one of s or t, but not both. In this article, we explore the existence for fixed numbers n = 2 and m = n - 1 of an optimal connected (n, m)-graph Gn,m for split reliability, that is, a connected graph with n vertices and m edges for which for any other such graph H, the split reliability of Gn,m is at least as large as that of H, for all values of p ? [0, 1]. Unlike the similar problems for all-terminal and two-terminal reliability, where only partial results are known, we completely solve the issue for split reliability, where we show that there is an optimal (n, m)-graph for split reliability if and only if n = 3, m = n -1, or n=m=4.