Cops and robber on variants of retracts and subdivisions of oriented graphs

TL;DR

This paper investigates pursuit-evasion games on directed graphs, analyzing how different move rules and graph modifications like retracts and subdivisions affect the cop number, and demonstrates the computational complexity persists in certain graph classes.

Contribution

It introduces and studies three variants of Cops and Robber on oriented graphs, analyzing invariance of cop number under retracts and subdivisions, and establishes complexity results.

Findings

Strong and normal cop numbers are invariant under specific retracts.

All variants remain computationally hard on 2-degenerate bipartite graphs.

The study extends understanding of pursuit-evasion dynamics on directed graph structures.

Abstract

\textsc{Cops and Robber} is one of the most studied two-player pursuit-evasion games played on graphs, where multiple \textit{cops}, controlled by one player, pursue a single \textit{robber}. The main parameter of interest is the \textit{cop number} of a graph, which is the minimum number of cops that can ensure the \textit{capture} of the robber. \textsc{Cops and Robber} is also well-studied on directed/oriented graphs. In directed graphs, two kinds of moves are defined for players: \textit{strong move}, where a player can move both along and against the orientation of an arc to an adjacent vertex; and \textit{weak move}, where a player can only move along the orientation of an arc to an \textit{out-neighbor}. We study three variants of \textsc{Cops and Robber} on oriented graphs: \textit{strong cop model}, where the cops can make strong moves while the robber can only make weak moves; \textit{normal cop model}, where both cops and the robber can only make weak moves; and \textit{weak cop model}, where the cops can make weak moves while the robber can make strong moves. We study the cop number of these models with respect to several variants of retracts on oriented graphs and establish that the strong and normal cop number of an oriented graph remains invariant in their strong and distributed retracts, respectively. Next, we go on to study all three variants with respect to the subdivisions of graphs and oriented graphs. Finally, we establish that all these variants remain computationally difficult even when restricted to the class of 2-degenerate bipartite graphs.