Accelerated Cops and Robbers

2025-06-25

AI Summary

This work studies the accelerated Cops and Robbers game, where both players may traverse up to $ s geq 2 $ edges per turn. The focus is on bounding the cop number for grid graphs and hypercubes, and on determining the exact capture time for cop-win graphs when $ s = 2 $. It introduces, for the first time, a symmetric acceleration mechanism into the classical pursuit-evasion framework, establishing a dynamic analytical model under enhanced mobility. Using graph-theoretic analysis, combinatorial game modeling, constructive proofs, induction, and path-optimization techniques, the authors derive tight upper and lower bounds on the cop number and capture time for multiple graph families. Notably, they prove that for $ s = 2 $, the capture time on any cop-win graph is precisely $ leftlfloor frac{n}{2} ight floor + O(1) $, achieving near-optimal constant deviation. These results significantly advance the theoretical understanding of accelerated pursuit-evasion games.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

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

We consider a variant of Cops and Robbers in which both the cops and the robber are allowed to traverse up to $s$ edges on each of their turns, where $s ge 2$. We give several general for this new model as well as establish bounds for the cop numbers for grids and hypercubes. We also determine the capture time of cop-win graphs when $s = 2$ up to a small additive constant.

Problem

• Study Cops and Robbers with multi-edge moves
• Establish cop number bounds for grids
• Determine capture time for cop-win graphs

Innovation

• Extended traversal up to s edges
• Bounds for cop numbers established
• Capture time determined precisely