Small Connected Planar Graph with 1-Cop-Move Number 4

TL;DR

This paper constructs a small connected planar graph with 720 vertices demonstrating that the 1-cop-move number can be between 4 and 7, improving previous bounds and showing 4 cops are sometimes barely sufficient.

Contribution

It presents a new small connected planar graph with a 1-cop-move number between 4 and 7, refining bounds established in prior work.

Findings

Constructed a 720-vertex connected planar graph with cop1 between 4 and 7.

Identified a subgraph requiring exactly 4 cops for the 1-cop-move game.

Improves previous bounds on cop numbers for planar graphs.

Abstract

This paper describes a 720-vertex connected planar graph G such that cop1(G), denoting the minimum number of cops needed to catch the robber in the 1-cop-move game on G, is at least 4 and at most 7. Furthermore, G has a connected subgraph H such that cop1(H) is exactly 4, meaning that 4 cops are barely sufficient to catch the robber in the 1-cop-move game on H. This is a significant improvement over the graph given by Gao and Yang in 2017.