The node cop‐win reliability of unicyclic and bicyclic graphs

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. We introduce a strengthening of one of these models by considering the probability that the remaining graph is cop-win. A graph is cop-win if one cop can win the game Cops and Robber. More precisely, for a graph G with nodes that are operational independently with probability p, the node cop-win reliability of G, denoted 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 NCRel(G, p) ≥ NCRel(H, p) for all p ∈ [0, 1] and all graphs H with n nodes and m edges. We show that such graphs exist among unicyclic and bicyclic graphs, respectively.