The node cop-win reliability of unicyclic and bicyclic graphs

TL;DR

This paper introduces a new network reliability measure based on the cop-win property from pursuit-evasion games, and identifies the most reliable unicyclic and bicyclic graphs under this measure.

Contribution

It defines the node cop-win reliability metric and proves the existence of uniformly most reliable graphs for unicyclic and bicyclic classes.

Findings

Uniformly most reliable unicyclic graphs identified.

Uniformly most reliable bicyclic graphs identified.

Contrast with non-existence of sparse graphs maximizing traditional reliability.

Abstract

Various models to quantify the reliability of a network have been studied where certain components of the graph may fail at random and the probability that the remaining graph is connected is the proxy for reliability. In this work we introduce a strengthening of one of these models by considering the probability that the remaining graph is not just connected but also cop-win. A graph is cop-win if one cop can guarantee capture of a fleeing robber in the well-studied pursuit-evasion game of Cops and Robber. More precisely, for a graph $G$ with nodes that are operational independently with probability $p$ and edges that are operational if and only if both of their endpoints are operational, the node cop-win reliability of $G$, denoted $\text{NCRel}(G,p)$, is the probability that the operational nodes induce a cop-win subgraph of $G$. It is then of interest to find graphs $G$ with $n$ nodes and $m$ edges such that $\text{NCRel}(G,p)\ge\text{NCRel}(H,p)$ for all $p\in[0,1]$ and all graphs $H$ with $n$ nodes and $m$ edges. Such a graph is called uniformly most reliable. We show that uniformly most reliable graphs exist for unicyclic and bicyclic graphs, respectively. This is in contrast to the fact that there are no known sparse graphs maximizing the corresponding notion of node reliability.