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

TL;DR

This paper constructs a specific graph with four independent vertices that has a cop number of three, providing counterexamples to existing conjectures and extending understanding of cop numbers in graph theory.

Contribution

It presents a counterexample to a conjecture about cop numbers in $C_{ ext{ell}}$-free graphs and generalizes bounds for cop numbers in classes defined by forbidden independent sets.

Findings

Counterexample to Sivaraman's conjecture on $C_{ ext{ell}}$-free graphs.

Proved bounds for cop numbers in graphs forbidding $pK_1$ and $qK_2$.

Introduced upper and lower threshold degrees for analyzing cop numbers.

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.