4K_1 free graphs on 13 vertices have cop number at most 2

TL;DR

This paper proves that all graphs without an induced 4K1 subgraph on 13 vertices or fewer have a cop number of at most 2, advancing understanding of the cops and robbers game on restricted graph families.

Contribution

It establishes an upper bound of 2 on the cop number for all graphs in the family forbidding 4K1 as an induced subgraph with up to 13 vertices, improving previous bounds.

Findings

Graphs in orb(4K_1) with up to 13 vertices have cop number at most 2.

The result narrows the search for potential counterexamples to a conjecture.

Supports the conjecture that smaller graphs in this family cannot have high cop number.

Abstract

The game of cops and robber has been studied for many years. Denoting $\mathsf{Forb}(4K_1)$ to be the family of all graphs that contain no induced subgraph isomorphic to $4K_1$ (e.g., with independence number less than $4$), we prove that for any $G\in\mathsf{Forb}(4K_1)$, we have $c(G)\leq 2$, where $c(\cdot)$ is the cop number. This improves a lower bound of a question proposed by Char et al. in a recent paper (arxiv, 2025), that any counterexample of a conjecture raised by Turcotte (2022) when $p=4$ must have at least 14 vertices.