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Guarding isometric subgraphs and Cops and Robber in planar graphs

Sebastián González Hermosillo de la Maza, Bojan Mohar

TL;DR

The paper addresses how to guard isometric subgraphs and bound robber capture in planar graphs under restricted cop movement. It introduces the wide shadow concept and a Helly-graph characterization, enabling a complete description of which subgraphs are 1-guardable, and uses this framework to derive a planar-graph bound $c_2(G) \le 3$ for the at-most-two-move variant. The key contributions are the equivalence: an isometric subgraph $H$ is $1$-guardable in any ambient graph iff $H$ is a Helly graph, and a constructive three-cop strategy proving Yang’s conjecture for planar graphs. The results link guardable subgraphs to dismantlable graphs and absolute retracts, offering structural tools for studying lazy-cop variants and extending classical planar cop-number bounds. This advances understanding of cop numbers in planar graphs and provides a unified approach via wide shadows and bypaths.

Abstract

In the game of Cops and Robbers, one of the most useful results is that an isometric path in a graph can be guarded by one cop. In this paper, we introduce the concept of wide shadow in a subgraph, and use it to characterize all 1-guardable graphs. As an application, we show that 3 cops can capture a robber in any planar graph with the added restriction that at most two cops can move simultaneously, proving a conjecture of Yang and strengthening a classical result of Aigner and Fromme.

Guarding isometric subgraphs and Cops and Robber in planar graphs

TL;DR

The paper addresses how to guard isometric subgraphs and bound robber capture in planar graphs under restricted cop movement. It introduces the wide shadow concept and a Helly-graph characterization, enabling a complete description of which subgraphs are 1-guardable, and uses this framework to derive a planar-graph bound for the at-most-two-move variant. The key contributions are the equivalence: an isometric subgraph is -guardable in any ambient graph iff is a Helly graph, and a constructive three-cop strategy proving Yang’s conjecture for planar graphs. The results link guardable subgraphs to dismantlable graphs and absolute retracts, offering structural tools for studying lazy-cop variants and extending classical planar cop-number bounds. This advances understanding of cop numbers in planar graphs and provides a unified approach via wide shadows and bypaths.

Abstract

In the game of Cops and Robbers, one of the most useful results is that an isometric path in a graph can be guarded by one cop. In this paper, we introduce the concept of wide shadow in a subgraph, and use it to characterize all 1-guardable graphs. As an application, we show that 3 cops can capture a robber in any planar graph with the added restriction that at most two cops can move simultaneously, proving a conjecture of Yang and strengthening a classical result of Aigner and Fromme.
Paper Structure (5 sections, 12 theorems, 3 equations, 3 figures)

This paper contains 5 sections, 12 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic notation
  3. Wide shadow and Helly graphs
  4. Other ways to guard a subgraph
  5. The 2-cop-move number of planar graphs

Key Result

Theorem 2.1

Every isometric path is $1$-guardable.

Figures (3)

  • Figure 1: The path $P$ consisting of thick edges is isometric in $G$. The vertex $y$ is the shadow of the vertex $v$ with respect to $s$. The three black vertices form the wide shadow of $v$.
  • Figure 2: The path $B = v_2b_3b_4v_5$ is a bypath of $P = v_1v_2v_3v_4v_5$, while $v_3xyv_5$ is not.
  • Figure 3: The subgraph $H=G-x$ induced by the square vertices is dismantlable but cannot be guarded by a single cop.

Theorems & Definitions (21)

  • Theorem 2.1: AignerDAM1984
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6: HellOLS2004
  • Theorem 2.7: BandeltJCTB1991
  • Lemma 2.8
  • ...and 11 more