Primal-Dual Cops and Robber

TL;DR

This paper introduces a novel variant of the Cops and Robber game on planar graphs, analyzing how the cops' movement between faces affects their ability to capture the robber depending on the graph's maximum degree.

Contribution

It establishes a precise condition on maximum degree for the cops to guarantee capture in this face-movement variant of the game.

Findings

A constant number of cops suffices for graphs with maximum degree ≤ 4.

Cops cannot guarantee capture on graphs with maximum degree > 4.

The result characterizes the relationship between graph degree and game outcome.

Abstract

Cops and Robber is a family of two-player games played on graphs in which one player controls a number of cops and the other player controls a robber. In alternating turns, each player moves (all) their figures. The cops try to capture the robber while the latter tries to flee indefinitely. In this paper we consider a variant of the game played on a planar graph where the robber moves between adjacent vertices while the cops move between adjacent faces. The cops capture the robber if they occupy all incident faces. We prove that a constant number of cops suffices to capture the robber on any planar graph of maximum degree $\Delta$ if and only if $\Delta \leq 4$.