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

TL;DR

This work proves that every $4K_1$-free graph on $13$ vertices satisfies $c(G)\le 2$, extending the understanding of cop numbers in $4K_1$-free families. The authors fix a vertex $u$ of maximum degree and study the induced subgraph $H=G-N[u]$ to show that if $H$ contains any of $K_5+K_1$, $K_4+K_2$, $K_3+K_3$, or $B_6$, then a two-cop strategy exists; otherwise, $H$ must be isomorphic to $B_6$ or the triangular prism $T_6$, which are handled by a careful case analysis. They treat all remaining configurations via computer-assisted verification, partitioning potential cases into Types A–D and exhaustively checking that $c(G)\le 2$ holds in every instance. The combination of structural lemmas with computational enumeration yields a complete verification for graphs on 13 vertices, advancing the threshold in Char et al.’s question and highlighting a robust approach for cop-number problems in constrained graph families.

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).

Figures (10)

• Figure 1: The graph $B_6$.
• Figure 2: The triangular prism $T_6$.
• Figure 3: The graph $H$ with a $K_4$ on $\{v,w,x,y\}$ and an edge $(s,t)$.
• Figure 4: The graph $H$ with the non‑adjacent crossing edges $(v,s)$ and $(w,t)$ added.
• Figure 5: The graph $H$ when $H[S_1] \cong H[S_2] \cong K_3$ and $|E(S_1, S_2)| = 2$.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 7
• Lemma 8
• Lemma 9
• proof