Limited Visibility Cops and Robbers

TL;DR

This paper studies a variation of the Cops and Robber game where visibility is limited to a fixed distance, analyzing strategies, resource requirements, and characterizing trees where a fixed number of cops suffice for all visibility ranges.

Contribution

It introduces a non-monotonic visibility model in Cops and Robbers, analyzes strategies across different graph families, and characterizes trees with universal cop numbers for all visibility ranges.

Findings

Visibility model is non-monotonic unlike traditional models.

Resource-intensive phases depend on graph structure.

Characterization of trees with fixed cop numbers for all visibility ranges.

Abstract

We consider a variation of the Cops and Robber game where the cops can only see the robber when the distance between them is at most a fixed parameter $\ell$. We consider the basic consequences of this definition for some simple graph families, and show that this model is not monotonic, unlike common models where the robber is invisible. We see that cops' strategy consists of a phase in which they need to "see" the robber (move within distance $\ell$ of the robber), followed by a phase in which they capture the robber. In some graphs the first phase is the most resource intensive phase (in terms of number of cops needed), while in other graphs, it is the second phase. Finally, we characterize those trees for which $k$ cops are sufficient to guarantee capture of the robber for all $\ell \ge 1$.