  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 ({\em
all-terminal reliability}) or that two specific vertices (or {\em terminals})
can communicate with each other ({\em two-terminal reliability}). A relatively
new measure is {\em 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 paper, we explore the existence for fixed numbers
$n \geq 2$ and $m \geq n-1$ of an {\em optimal} connected $(n,m)$-graph
$G_{n,m}$ 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
$G_{n,m}$ is at least as large as that of $H$, for {\em all} values of $p \in
[0,1]$. 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 $n\leq 3$, $m=n-1$, or
$n=m=4$.
