Guarding isometric subgraphs and Cops and Robber in planar graphs

TL;DR

This paper introduces the concept of wide shadow to characterize 1-guardable graphs and proves that three cops can capture a robber in any planar graph with movement restrictions, confirming a conjecture and strengthening classical results.

Contribution

It presents a new characterization of 1-guardable graphs using wide shadows and proves a novel result on cop number in planar graphs under movement constraints.

Findings

All 1-guardable graphs characterized by wide shadows.

Three cops suffice to capture a robber in planar graphs with movement restrictions.

Confirmed a conjecture of Yang and strengthened classical results of Aigner and Fromme.

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.