On the minimum order of k-cop-win graphs

TL;DR

This paper determines the smallest connected graphs with cop number 3, establishing the Petersen graph as unique at this order, and explores the relationship between minimum order and Meyniel's conjecture.

Contribution

It proves the minimum order for connected graphs with cop number 3 is 10 and identifies the Petersen graph as the unique such graph, also relating this to Meyniel's conjecture.

Findings

Minimum order of a connected graph with cop number 3 is 10.

The Petersen graph is the unique graph with this property.

Computational results for graphs up to order 10.

Abstract

We consider the minimum order graphs with a given cop number. We prove that the minimum order of a connected graph with cop number 3 is 10, and show that the Petersen graph is the unique isomorphism type of graph with this property. We provide the results of a computational search on the cop number of all graphs up to and including order 10. A relationship is presented between the minimum order of graph with cop number $k$ and Meyniel's conjecture on the asymptotic maximum value of the cop number of a connected graph.