The Cops and Robber game on graphs with forbidden (induced) subgraphs

TL;DR

This paper investigates the cop number in graphs with forbidden subgraphs or induced subgraphs, providing characterizations for graphs with bounded cop number and relating it to tree-width.

Contribution

It offers a complete characterization of graphs with bounded cop number under forbidden subgraph conditions and connects cop number bounds to tree-width.

Findings

Characterization of graphs with bounded cop number under forbidden subgraph conditions

Bound on cop number in terms of tree-width

Analysis of cop number in classes of graphs defined by forbidden induced subgraphs

Abstract

The two-player, complete information game of Cops and Robber is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if, after a move, a cop and the robber are on the same vertex. The minimum number of cops needed to catch the robber on a graph is called the cop number of that graph. In this paper, we study the cop number in the classes of graphs defined by forbidding one or more graphs as either subgraphs or induced subgraphs. In the case of a single forbidden graph we completely characterize (for both relations) the graphs which force bounded cop number. En passant, we bound the cop number in terms of tree-width.