On the Conjecture of the Smallest 3-Cop-Win Planar Graph

Abstract

In the game of Cops and Robbers on a graph \(G=(V,E)\), k cops try to catch a robber. The minimum number of cops required to win is called the cop number, denoted by c(G). For a planar graph G, it is known that \(c(G)\le 3\). It is a conjecture that the regular dodecahedral graph of order 20 is the smallest planar graph whose cop number is three. As the very first attack on this conjecture, we provide the following evidences in this paper: (1) any planar graph of order at most 19 has the winning vertex at which two cops can capture the robber, and (2) a special planar graph of order 19 that is constructed from the regular dodecahedral graph has the cop number of two.