A cop-robber game on metric graphs

TL;DR

The paper studies a continuous-time cop–robber pursuit on compact metric graphs, showing that a finite critical speed exists for every connected graph. It develops structural lemmas (e.g., edge-length shortening invariance and scaling) and analyzes star graphs to derive explicit bounds, including s^*(S_k) = (2k-3)^+. The authors prove the existence of optimal strategies with a maximal-speed property and provide proofs for finiteness and strategy structure, extending to more complex graphs such as combs. This work advances understanding of pursuit–evasion on networks by combining geometric graph analysis with differential-game ideas and fixed-preference strategies under incomplete information.

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.

Figures (6)

• Figure 1: The $k$-star graph $S_k$ (for $k=5$): a tree with one central node and $k$ leaves.
• Figure 2: The graph $B_k$ (for $k=5$): a path of $k$ vertices, each with a leaf attached.
• Figure 3: Graphs $G$ and $G'$, where $G'$ is obtained from $G$ by shortening the edge from $v_1$ to the leaf $v_2$.
• Figure 4: A single edge is cleaned in Phase 1.
• Figure 5: Expansion of free zones during one iteration.

Theorems & Definitions (15)

• Theorem 2.1
• Theorem 2.2
• Conjecture 2.3
• Proposition 2.4
• Theorem 2.5
• Remark 2.6
• Theorem 2.7
• Remark 2.8
• Lemma 3.1
• Remark 3.2