A cop-robber game on metric graphs

TL;DR

This paper investigates a continuous version of the cop-robber game on metric graphs, establishing conditions under which the cop can guarantee capture based on their relative speeds.

Contribution

It introduces a new variant of the cop-robber game on metric graphs and proves the existence of a speed ratio ensuring the cop's guaranteed capture.

Findings

Existence of a speed ratio s > 0 for guaranteed capture on any compact metric graph.

The cop can guarantee capture if their speed exceeds s times the robber's speed.

The game extends classical discrete models to continuous metric graphs.

Abstract

We study a variant of the classical cop-robber game played on compact metric graphs, where each edge is assigned a positive length and identified with a real interval of corresponding length. In this setting, both the cop and the robber move continuously along the edges, subject to upper bounds on their speeds. The cop has no knowledge of the robber's location and must choose a continuous path through the graph that is guaranteed to intersect the robber's trajectory at some point in time. We show that for every compact metric graph, there exists a constant s > 0 such that if the cop's speed exceeds s times the robber's speed, then the cop can guarantee capture.