Cops and Robbers playing on edges

Abstract

In the game of cops and robbers, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. In this paper, we consider the variant of the game in which both players play on edges instead of vertices, yielding the edge cop number. We relate the new graph parameter to the classic one, investigate Meyniel extremal families, and characterize edge cop-win graphs. We also play the game on random graphs and planar graphs.