Guarding isometric subgraphs and Lazy Cops and Robbers

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 on a graph, and use it to provide a short proof of the characterization
of $1$-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
strenghtening a classical result by Aigner and Fromme.