Cops and robber in graphs with bounded vertex cover number

2026-02-07

AI Summary

This study addresses the problem of bounding the cop number in connected graphs with vertex cover number $k$, offering a structural perspective toward Meyniel’s conjecture. By integrating vertex cover structure analysis, combinatorial optimization, and asymptotic methods, the work establishes the first sublinear upper bound on the cop number parameterized solely by $k$. Specifically, it proves that any connected graph with vertex cover number $k$ has cop number at most $k / 2^{(1 - o(1))\sqrt{\log k}}$. This result breaks away from the conventional framework that depends on the total number of vertices $n$, providing new evidence for Meyniel’s conjecture within structurally restricted graph classes.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Meyniel's conjecture states that $n$-vertex connected graphs have cop number $O(\sqrt{n})$. The current best known upper bound is $n/2^{(1-o(1))\sqrt{\log n}}$, proved independently by Lu and Peng (2011), and by Scott and Sudakov (2011). In this paper, we extend their result by showing that every connected graph with vertex cover number $k$ has cop number at most $k/2^{(1-o(1))\sqrt{\log k}}$. This is the first sublinear upper bound on the cop number in terms of the vertex cover number.

Problem

• cop number
• vertex cover number
• Meyniel's conjecture
• graph searching
• bounded parameter

Innovation

• cop number
• vertex cover number
• sublinear upper bound
• Meyniel's conjecture
• graph searching