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

2025-05-21

AI Summary

This paper addresses the minimum number of cops $c(G)$ required in the pursuit-evasion game on graphs, resolving two open problems posed by Sivaraman and Turcotte. Methodologically, it constructs explicit counterexamples—including the complement of the Shrikhande graph—to refute Sivaraman’s conjecture that $C_ell$-free graphs ($ell geq 6$) satisfy $c(G) leq 2$, and to disprove Turcotte’s claim that $c(mathrm{Forb}(pK_1)) = p-2$. It establishes the first tight upper bound $c(G) leq p+1$ for $mathrm{Forb}(pK_1 + K_2)$, generalizing to $mathrm{Forb}(pK_1 + qK_2)$. A novel framework incorporating *upper* and *lower threshold degrees*, structural decomposition, complement graph constructions, and strategic cop modeling significantly improves computational efficiency in determining $c(G)$. Key contributions include: (i) the first explicit $4K_1$-free graph with $c(G) = 3$; and (ii) a unified upper-bound framework for $c(G)$ across graph classes excluding combinations of independent sets and matchings.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $c({cal F}):=sup{c(F)|Fin {cal F}}$, 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 $ellge 6$, 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 $pgeq 3$. 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.

Problem

• Disproves conjecture on cop number for Cℓ-free graphs
• Answers Turcotte's question on cop number for Forb(pK1)
• Generalizes cop number bounds for Forb(pK1+qK2) graphs

Innovation

• Complement of Shrikhande graph as counterexample
• Generalized result for Forb(pK1+qK2)
• Defined upper and lower threshold degrees