  We consider a variant of the game of Cops and Robbers, called Containment, in
which cops move from edge to adjacent edge, the robber moves from vertex to
adjacent vertex (but cannot move along an edge occupied by a cop). The cops win
by "containing" the robber, that is, by occupying all edges incident with a
vertex occupied by the robber. The minimum number of cops, $\xi(G)$, required
to contain a robber played on a graph $G$ is called the containability number,
a natural counterpart of the well-known cop number $c(G)$. This variant of the
game was recently introduced by Komarov and Mackey, who proved that for every
graph $G$, $c(G) \le \xi(G) \le \gamma(G) \Delta(G)$, where $\gamma(G)$ and
$\Delta(G)$ are the domination number and the maximum degree of $G$,
respectively. They conjecture that an upper bound can be improved and, in fact,
$\xi(G) \le c(G) \Delta(G)$. (Observe that, trivially, $c(G) \le \gamma(G)$.)
This seems to be the main question for this game at the moment. By
investigating expansion properties, we provide asymptotically almost sure
bounds on the containability number of binomial random graphs $G(n,p)$ for a
wide range of $p=p(n)$, showing that it forms an intriguing zigzag shape. This
result also proves that the conjecture holds for some range of $p$ (or holds up
to a constant or an $O(\log n)$ multiplicative factors for some other ranges).
