Existence of Optimally-Greatest Digraphs for Strongly Connected Node Reliability

TL;DR

This paper investigates the existence and properties of optimally-greatest directed graphs for strong connectivity reliability, especially circulant graphs, under various node operational probabilities.

Contribution

It introduces a new model for network reliability with node failures and characterizes optimally-greatest circulant graphs for different operational probabilities.

Findings

Existence of optimally-greatest digraphs depends on the number of arcs.

Identifies specific circulant graphs that are optimally-greatest near zero and near one probability.

Provides conditions on the number of vertices for certain circulant graphs to be optimal.

Abstract

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.