  We consider a variant of the game of Cops and Robbers, called Lazy Cops and
Robbers, where at most one cop can move in any round. We investigate the
analogue of the cop number for this game, which we call the lazy cop number.
Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who
provided asymptotic upper and lower bounds on the lazy cop number of the
hypercube. By investigating expansion properties, we provide asymptotically
almost sure bounds on the lazy cop number of binomial random graphs
$\mathcal{G}(n,p)$ for a wide range of $p=p(n)$. By coupling the probabilistic
method with a potential function argument, we also improve on the existing
lower bounds for the lazy cop number of hypercubes. Finally, we provide an
upper bound for the lazy cop number of graphs with genus $g$ by using the
Gilbert-Hutchinson-Tarjan separator theorem.
