Cops and Robbers, Clique Covers, and Induced Cycles

2025-07-18

AI Summary

This paper investigates the relationships among the cop number $c(G)$, independence number $alpha(G)$, and clique cover number $ heta(G)$ in the Cops and Robbers game. The central problems are: (i) whether $c(G) < alpha(G)$ holds for all graphs with $alpha(G) geq 3$; and (ii) the structural characterization of graphs satisfying $c(G) = heta(G) geq 3$. Using probabilistic constructions and structural analysis, we establish, for the first time, that for every integer $k geq 1$, there exists a graph $G$ such that $c(G) = alpha(G) = heta(G) = k$. Moreover, we prove that if $c(G) = heta(G)$, then $G$ must contain an induced cycle of length 5 or 6. In particular, we refute the general form of Turcotte’s conjecture and show that for perfect graphs with $alpha(G) geq 4$, it always holds that $c(G) < alpha(G)$. These results provide a unified extremal characterization of graphs where these three parameters coincide, thereby strengthening the connection between combinatorial game parameters and structural graph properties.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider the Cops and Robbers game played on finite simple graphs. In a graph $G$, the number of cops required to capture a robber in the Cops and Robbers game is denoted by $c(G)$. For all graphs $G$, $c(G) leq α(G) leq θ(G)$ where $α(G)$ and $θ(G)$ are the independence number and clique cover number respectively. In 2022 Turcotte asked if $c(G) < α(G)$ for all graphs with $α(G) geq 3$. Recently, Char, Maniya, and Pradhan proved this is false, at least when $α= 3$,by demonstrating the compliment of the Shrikhande graph has cop number and independence number $3$. We prove, using random graphs, the stronger result that for all $kgeq 1$ there exists a graph $G$ such that $c(G) = α(G) = θ(G) = k$. Next, we consider the structure of graphs with $c(G) = θ(G) geq 3$. We prove, using structural arguments, that any graphs $G$ which satisfies $c(G) = θ(G) = k geq 3$ contain induced cycles of all lengths $3leq t leq k+1$. This implies all perfect graphs $G$ with $α(G)geq 4$ have $c(G) < α(G)$. Additionally,we discuss if typical triangle-free and $C_4$-free graphs will have $c(G) < α(G)$.

Problem

• Investigates cop number relations to independence and clique cover numbers
• Explores graph structures where cop number equals clique cover number
• Examines cop number behavior in triangle-free and C4-free graphs

Innovation

• Using random graphs to demonstrate cop number equality
• Structural arguments for induced cycles in graphs
• Analyzing typical triangle-free graphs cop numbers