Cop numbers for subclasses of partial cubes

Abstract

The game of Cops and Robbers is a classical pursuit--evasion game on graphs. For a graph $G$, the cop number $c(G)$ is the minimum number of cops needed to guarantee the capture of a robber on $G$. Although this parameter has been determined for several fundamental graph classes, comparatively few exact results are known for partial cubes and their subclasses. We first establish an upper bound for every finite median graph $M$ in terms of its tree-dimension, which improves Crawford and Ir\v{s}i\v{c} Chenoweth's bound significantly. This result refines the previous upper bound expressed in terms of a hypercube embedding dimension and can give a substantially smaller estimate. Then we investigate the cop numbers of simplex graphs---a subclass of partial cubes. For a finite graph $G$, the simplex graph $S(G)$ has the cliques of $G$, including the empty clique, as its vertices, with two cliques adjacent whenever they differ in exactly one vertex. We establish a general lower bound for $c(S(G))$ in terms of the clique number of $G$ and a general upper bound in terms of its chromatic number. Finally, as direct applications, we determine the exact values of cop numbers of some special simplex graphs---bipartite wheels, Fibonacci and Lucas cubes.