Existence of Optimally-Greatest Digraphs for Strongly Connected Node Reliability

In this paper, we introduce a new model to study network reliability with node failures. This model, strongly connected node reliability, is the directed variant of node reliability and measures the probability that the operational vertices induce a subdigraph that is strongly connected. If we are restricted to directed graphs with $n$ vertices and $n+1\leq m\leq 2n-3$ or $m=2n$ arcs, an optimally-greatest digraph does not exist. Furthermore, we study optimally-greatest directed circulant graphs when the vertices operate with probability $p$ near zero and near one. In particular, we show that the graph $\Gamma\left(\mathbb{Z}_n,\{1,-1\}\right)$ is optimally-greatest for values of $p$ near zero. Then, we determine that the graph $\Gamma\left(\mathbb{Z}_{n},\{1,\frac{n+2}{2}\}\right)$ is optimally-greatest for values of $p$ near one when $n$ is even. Next, we show that the graph $\Gamma\left(\mathbb{Z}_{n},\{1,2(3^{-1})\}\right)$ is optimally-greatest for values of $p$ near one when $n$ is odd and not divisible by three and that $\Gamma\left(\mathbb{Z}_{n},\{1,3(2^{-1})\}\right)$ is optimally-greatest for values of $p$ near one when $n$ is odd and divisible by three. We conclude with a discussion of open problems.