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

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. cL(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) ≤ 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 3×3 rooks graph (K3 K3) 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.