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$$ p $$. 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$$ s $$ and t$$ t $$, we consider the probability that every vertex communicates with one of s$$ s $$ or t$$ t $$, but not both. In this article, we explore the existence for fixed numbers n≥2$$ n\ge 2 $$ and m≥n−1$$ m\ge n-1 $$ of an optimal connected (n,m)$$ \left(n,m\right) $$‐graph Gn,m$$ {G}_{n,m} $$ for split reliability, that is, a connected graph with n$$ n $$ vertices and m$$ m $$ edges for which for any other such graph H$$ H $$, the split reliability of Gn,m$$ {G}_{n,m} $$ is at least as large as that of H$$ H $$, for all values of p∈[0,1]$$ p\in \left[0,1\right] $$. 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)$$ \left(n,m\right) $$‐graph for split reliability if and only if n≤3$$ n\le 3 $$, m=n−1$$ m=n-1 $$, or n=m=4$$ n=m=4 $$.