Cops and Robber Game with a Fast Robber on Expander Graphs and Random Graphs

TL;DR

This paper studies a variant of the Cops and Robber game where the robber can move arbitrarily fast but cannot pass through cops, providing characterizations, algorithms, and bounds for different graph classes, including random and expander graphs.

Contribution

It characterizes graphs with a single cop needed, provides an algorithm for detection, and establishes bounds for random, regular, and product graphs in this game variant.

Findings

Graphs with one cop characterized and detected efficiently.

Expander graphs require a number of cops proportional to their properties.

Random and regular graphs have bounds on cops needed, depending on their structure.

Abstract

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e. can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c_{infty}(G) denote the number of cops needed to capture the robber in a graph G in this variant. We characterize graphs G with c_{infty}(G)=1, and give an O(|V(G)|^2) algorithm for their detection. We prove a lower bound for c_{infty} of expander graphs, and use it to prove three things. The first is that if np > 4.2 log n then the random graph G = G(n,p) asymptotically almost surely has e1/p < c_{infty}(G) < e2 log (np)/p, for suitable constants e1 and e2. The second is that a fixed-degree random regular graph G with n vertices asymptotically almost surely has c_{infty}(G) = Theta(n). The third is that if G is a Cartesian product of m paths, then n / 4km^2 < c_{infty}(G) < n / k, where n=|V(G)| and k is the number of vertices of the longest path.