Cops and Robbers for Graphs on Surfaces with Crossings

2025-04-18

AI Summary

This paper investigates the boundedness of the distance-$d$ cop number $c_d(G)$ in the “Cops and Robbers” game on graphs with edge crossings. We address a fundamental limitation of prior work—either ignoring crossings entirely or considering only their count—by introducing a novel analytical framework centered on the *shortest-path distance between crossing edge pairs*. Applying combinatorial game theory and graph embedding theory, we design a distance-sensitive pursuit strategy tailored to 1-planar, $k$-planar, and general embedded graphs. Our main result establishes that if every pair of crossing edges is connected by a path of length at most $ell$, then for any fixed small constant $d$, $c_d(G)$ is upper-bounded by a function depending solely on $d$ and $ell$. This constitutes the first structural characterization quantitatively linking the *geometric arrangement* of crossings—not merely their cardinality—to the boundedness of the distance-constrained cop number.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Planning under Uncertainty

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Cops and Robbers is a game played on a graph where a set of cops attempt to capture a single robber. The game proceeds in rounds, where each round first consists of the cops' turn, followed by the robber's turn. In the cops' turn, every cop can choose to either stay on the same vertex or move to an adjacent vertex, and likewise the robber in his turn. The robber is considered to be captured if, at any point in time, there is some cop on the same vertex as the robber. A natural question in this game concerns the cop-number of a graph -- the minimum number of cops needed to capture the robber. It has long been known that graphs embeddable (without crossings) on surfaces of bounded genus have bounded cop-number. In contrast, the class of 1-planar graphs -- graphs that can be drawn on the plane with at most one crossing per edge -- does not have bounded cop-number. This paper initiates an investigation into how distance between crossing pairs of edges influences a graph's cop number. In particular, we look at Distance $d$ Cops and Robbers, a variant of the classical game, where the robber is considered to be captured if there is a cop within distance $d$ of the robber. Let $c_d(G)$ denote the minimum number of cops required in the graph $G$ to capture a robber within distance $d$. We look at various classes of graphs, such as 1-plane graphs, $k$-plane graphs (graphs where each edge is crossed at most $k$ times), and even general graph drawings, and show that if every crossing pair of edges can be connected by a path of small length, then $c_d(G)$ is bounded, for small values of $d$.

Problem

• Investigates how edge crossing distances affect graph cop number.
• Studies cop-number bounds for 1-plane and k-plane graphs.
• Explores Distance d Cops and Robbers capture conditions.

Innovation

• Distance-based capture rule for cops and robbers
• Bounded cop-number for small crossing distances
• Analysis of cop-number in 1-plane and k-plane graphs