Blind cop-width and balanced minors of graphs

TL;DR

This paper studies a pursuit-evasion game on graphs where cops are blind and can move instantly, introducing the concept of blind cop-width and linking it to graph parameters like treewidth, with new bounds involving balanced minors.

Contribution

It introduces the concept of blind cop-width and balanced minors, establishing new bounds relating pursuit-evasion parameters to graph minors and treewidth.

Findings

Blind cop-width is bounded below by treewidth.

Balanced minors have equal-sized branch sets.

New lower bounds relate pursuit parameters to graph minors.

Abstract

We investigate a pursuit-evasion game on an undirected graph in which a robber, moving at a fixed constant speed, attempts to evade a team of cops who are blind to the robber's location and can quickly travel between any pair of vertices in the graph. The blind cop-width is the minimum number of cops needed to catch the robber on a given graph. We link it with other known graph parameters defined in terms of pursuit-evasion games, and show a new lower bound with respect to treewidth. The proof introduces the notion of balanced minors, where all branch sets of a minor model have equal size.