The 3x3 rooks graph is the unique smallest graph with lazy cop number 3

TL;DR

This paper proves that the 3x3 rooks graph is the smallest graph requiring three lazy cops to catch a robber, establishing its uniqueness among nine-vertex graphs for this property.

Contribution

It establishes the 3x3 rooks graph as the unique smallest graph with lazy cop number 3, extending known results to lazy cops.

Findings

The 3x3 rooks graph has lazy cop number 3.

Graphs with fewer than nine vertices have lazy cop number at most 2.

Uniqueness of the 3x3 rooks graph for lazy cop number 3 is proven.

Abstract

In the ordinary version of the pursuit-evasion game "cops and robbers", a team of cops and a robber occupy vertices of a graph and alternately move along the graph's edges, with perfect information about each other. If a cop lands on the robber, the cops win; if the robber can evade the cops indefinitely, he wins. In the variant "lazy cops and robbers", the cops may only choose one member of their squad to make a move when it's their turn. The minimum number of cops (respectively lazy cops) required to catch the robber is called the "cop number" (resp. "lazy cop number") of G and is denoted $c(G)$ (resp. $c_L(G)$). Previous work by Beveridge at al. has shown that the Petersen graph is the unique graph on ten vertices with $c(G)=3$, and all graphs on nine or fewer vertices have $c(G)\leq 2$. (This was a self-contained mathematical proof of a result found by computational search by Baird and Bonato.) In this article, we prove a similar result for lazy cops, namely that the 3x3 rooks graph ($K_3\square K_3$) is the unique graph on nine vertices which requires three lazy cops, and a graph on eight or fewer vertices requires at most two lazy cops.