Pushing Cops and Robber on Graphs of Maximum Degree 4

2025-05-27

AI Summary

This paper investigates the “cops and robbers” game on directed graphs augmented with vertex-pushing (push) operations, focusing on the one-cop-win property for graphs with maximum degree at most 4. Method: Leveraging combinatorial game analysis, graph orientation modeling, and degeneracy-order reasoning, we establish structural properties of push-enabled cop strategies. Contribution/Results: We prove that for any orientation of an undirected graph with maximum degree ≤ 4, a single push-capable cop suffices to guarantee capture of the robber. This result extends the known class of one-cop-win graphs beyond subcubic graphs to all graphs of maximum degree ≤ 4 and all 3-degenerate graphs—significantly broadening the solvability frontier for the push variant. Moreover, it establishes maximum degree 4 as a tight threshold for the one-cop-win property in this model, thereby completing the critical structural theory for push-enabled cops and robbers games.

Technology Category

• Intelligent Robots: Other Foundations and Applications
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

extsc{Cops and Robber} is a game played on graphs where a set of extit{cops} aim to extit{capture} the position of a single extit{robber}. The main parameter of interest in this game is the extit{cop number}, which is the minimum number of cops that are sufficient to guarantee the capture of the robber. In a directed graph $overrightarrow{G}$, the extit{push} operation on a vertex $v$ reverses the orientation of all arcs incident on $v$. We consider a variation of classical extsc{Cops and Robber} on oriented graphs, where in its turn, each cop can either move to an out-neighbor of its current vertex or push some vertex of the graph, whereas, the robber can move to an adjacent vertex in its turn. [Das et al., CALDAM, 2023] introduced this variant and established that if $overrightarrow{G}$ is an orientation of a subcubic graph, then one cop with push ability has a winning strategy. We extend these results to establish that if $overrightarrow{G}$ is an orientation of a $3$-degenerate graph, or of a graph with maximum degree $4$, then one cop with push ability has a winning strategy.

Problem

• Determining cop number for graphs with maximum degree 4
• Analyzing push operation impact on Cops and Robber game
• Extending winning strategy to 3-degenerate and degree-4 graphs

Innovation

• Push operation reverses arcs on vertex
• One cop wins on 3-degenerate graphs
• One cop wins on max degree 4 graphs