Containment game played on random graphs: another zig-zag theorem

TL;DR

This paper studies a variant of the Cops and Robbers game called Containment on random graphs, providing bounds on the containability number and supporting a conjecture about its relation to the cop number.

Contribution

It introduces asymptotic bounds for the containability number on random graphs, confirming the conjecture in certain ranges of edge probability.

Findings

Containability number exhibits a zigzag pattern on $G(n,p)$.

The conjecture $\xi(G) \,\le c(G) \,\Delta(G)$ holds for some ranges of $p$.

Provides probabilistic bounds using expansion properties.

Abstract

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).