$4K_1$-free graph with the cop number $3$

TL;DR

This work advances the understanding of the cop number in forbidden-subgraph graph classes by constructing a $(4K_1, C_\ell)$-free graph with cop number $3$ via the complement of the Shrikhande graph, thereby giving a counterexample to a conjecture that $C_\ell$-free graphs with $\ell\ge6$ have cop number at most $2$. It establishes $c(\text{Forb}(4K_1))=3$ and introduces the concepts of upper and lower threshold degrees, showing $ut(\text{Forb}(4K_1))=6$ and $lt(\text{Forb}(4K_1))\ge3$, to aid computational determination of $c(\mathcal{G})$. The paper further provides general bounds for $c(\text{Forb}(pK_1+qK_2))$, proving $c(G)\le p+2q-2$ (and tighter forms in special cases), and discusses computational strategies that leverage threshold degrees. Collectively, these results illuminate the cop-number landscape in restricted graph families and offer practical tools for analyzing large classes of graphs.

Abstract

The game of cops and robber is a two-player turn-based game played on a graph where the cops try to capture the robber. The cop number of a graph $G$, denoted by $c(G)$ is the minimum number of cops required to capture the robber. For a given class of graphs ${\cal F}$, let $c({\cal F}):=\sup\{c(F)|F\in {\cal F}\}$, and let Forb$({\cal F})$ denote the class of ${\cal F}$-free graphs. We show that the complement of the Shrikhande graph is $(4K_1,C_{\ell}$)-free for any $\ell \geq 6$ and has the cop number~$3$. This provides a counterexample for the conjecture proposed by Sivaraman (arxiv, 2019) which states that if $G$ is $C_{\ell}$-free for all $\ell\ge 6$, then $c(G)\le 2$. This also gives a negative answer to the question posed by Turcotte (Discrete Math. 345:112660 (2022)) 112660. to check whether $c($Forb$(pK_1))=p-2$. Turcotte also posed the question to check whether $c($Forb$(pK_1+K_2))\leq p+1$, for $p\geq 3$. We prove that this result indeed holds. We also generalize this result for Forb$(pK_1+qK_2)$. Motivated by the results of Baird et al. (Contrib. Discrete Math. 9:70--84 (2014)) and Turcotte and Yvon (Discrete Appl. Math. 301:74--98 (2021)), we define the upper threshold degree and lower threshold degree for a particular class of graphs and show some computational advantage to find the cop number using these.

Figures (3)

• Figure 1: Some special graphs.
• Figure 2: (a) The Shrikhande graph and (b) The complement of the Shrikhande graph.
• Figure 3: (a) $\overline{C_6}$ and (b) $\overline{P_6}$

Theorems & Definitions (74)

• Theorem 1: Chudnovsky24
• Theorem 2
• Corollary 1
• Conjecture 1: Sivaraman19_1
• Theorem 3
• Theorem 4
• Lemma 1
• proof
• Claim 1.1
• proof : Proof of Claim \ref{['Shiraba+1bnotinK']}