On the cop number of generalized Petersen graphs

TL;DR

This paper establishes upper bounds on the cop number for generalized Petersen graphs and I-graphs using a novel covering space strategy, providing exact values for certain families.

Contribution

It introduces a new approach using infinite cyclic coverings to analyze cop numbers and determines exact cop numbers for specific families of generalized Petersen graphs.

Findings

Cop number of all generalized Petersen graphs is at most 4.

Finite isometric subtrees are 1-guardable.

Cop number of any connected I-graph is at most 5.

Abstract

We show that the cop number of every generalized Petersen graph is at most 4. The strategy is to play a modified game of cops and robbers on an infinite cyclic covering space where the objective is to capture the robber or force the robber towards an end of the infinite graph. We prove that finite isometric subtrees are 1-guardable and apply this to determine the exact cop number of some families of generalized Petersen graphs. We also extend these ideas to prove that the cop number of any connected I-graph is at most 5.