Further results on the Hunters and Rabbit game through monotonicity

The Hunters and Rabbit game is played on a graph G where the Hunter player shoots at k vertices in every round while the Rabbit player occupies an unknown vertex and, if not shot, must move to a neighbouring vertex. The Rabbit player wins if and only if it is not shot indefinitely. The hunter number $h\left(G\right)$ of a graph G is the minimum k such that the Hunter player has a winning strategy. We propose a notion of monotonicity, embodied in the monotone hunter number $mh\left(G\right)$, for this game imposing that a vertex that has already been shot “must not host the rabbit anymore”.

We show that $pw\left(G\right)\le mh\left(G\right)\le pw\left(G\right)+1$ for any graph G with pathwidth $pw\left(G\right)$, implying that computing, or even approximating, $mh\left(G\right)$ is NP-hard. Then, we show that $mh\left(G\right)$ can be computed in polynomial time in split graphs, interval graphs, cographs and trees. These results go through structural characterisations which relate the monotone hunter number with the pathwidth. In all these cases, we either specify the hunter number or show that there may be an arbitrary gap between h and mh, i.e., that monotonicity does not help. In particular, for every $k\ge 3$, we construct a tree T with $h\left(T\right)=2$ and $mh\left(T\right)=k$. We conclude by proving that computing h (resp., mh) is $\mathrm{FPT}$ parameterised by the vertex cover number.