Constructing Geometric Graphs of Cop Number Three

TL;DR

This paper introduces constructions of geometric graphs with cop number three, demonstrating the existence of infinitely many such graphs and exploring their properties through subdivision and clique substitution methods.

Contribution

It provides new methods to construct geometric graphs with cop number three, expanding understanding of pursuit games on geometric graph classes.

Findings

Infinite families of planar geometric graphs with cop number three are constructed.

Subdivision techniques preserve or increase the cop number in geometric graphs.

Clique substitution methods produce geometric graphs with cop number at least that of the original graphs.

Abstract

The game of cops and robbers is a pursuit game on graphs where a set of agents, called the cops try to get to the same position of another agent, called the robber. Cops and robbers has been studies on several classes of graphs including geometrically represented graphs. For example, it has been shown that string graphs, including geometric graphs, have cop number at most 15. On the other hand, little is known about geometric graphs of any cop number less than 15 and there is only one example of a geometric graph of cop number three that has as many as 1440 vertices. In this paper we present a construction for subdividing planar graphs of maximum degree $\le 5$ into geometric planar graphs of at least the same cop number. Indeed, our construction shows that there are infinitely many planar geometric graphs of cop number three. We also present another construction that consists in clique substitutions alongside subdividing the edges in a planar graph of maximum degree $\le 9$, resulting in geometric, but not necessarily planar, graphs of at least the same cop number as the starting graphs.