# Recontamination helps a lot to hunt a rabbit

Foivos Fioravantes, 9 Oct 2023

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 it is not shot, must move to a neighbouring vertex after each round. The Rabbit player wins if it can ensure that its position is never shot. The Hunter player wins otherwise. The hunter number \(\operatorname{h}(G)\) of a graph \(G\) is the minimum integer \(k\) such that the Hunter player has a winning strategy (i.e., allowing him to win whatever be the strategy of the Rabbit player). This game has been studied in several graph classes, in particular in bipartite graphs (grids, trees, hypercubes,…), but the computational complexity of computing \(\operatorname{h}(G)\) remains open in general graphs and even in more restricted graph classes such as trees. To progress further in this study, we propose a notion of monotonicity (a well-studied and useful property in classical pursuit-evasion games such as Graph Searching games) for the Hunters and Rabbit game imposing that, roughly, a vertex that has already been shot ``must not host the rabbit anymore’’. This allows us to obtain new results in various graph classes.

More precisely, let the monotone hunter number \(\operatorname{mh}(G)\) of a graph \(G\) be the minimum integer \(k\) such that the Hunter player has a monotone winning strategy. We show that \(\operatorname{pw}(G) \leq \operatorname{mh}(G) \leq \operatorname{pw}(G)+1\) for any graph \(G\) with pathwidth \(\operatorname{pw}(G)\), which implies that computing \(\operatorname{mh}(G)\), or even approximating \(\operatorname{mh}(G)\) up to an additive constant, is \(\mathsf{NP}\)-hard. Then, we show that \(\operatorname{mh}(G)\) can be computed in polynomial time in split graphs, interval graphs, cographs and trees. These results go through structural characterisations which allow us to relate the monotone hunter number with the pathwidth in some of these graph classes. In all cases, this allows us to specify the hunter number or to show that there may be an arbitrary gap between \(\operatorname{h}(G)\) and \(\operatorname{mh}(G)\), i.e., that monotonicity does not help. In particular, we show that, for every \(k\geq 3\), there exists a tree \(T\) with \(\operatorname{h}(T)=2\) and \(\operatorname{mh}(T)=k\). We conclude by proving that computing \(\operatorname{h}(G)\) (resp., \(\operatorname{mh}(G)\)) is \(\mathsf{FPT}\) parameterised by the minimum size of a vertex cover. This is a joined work with Thomas Dissaux, Harmender Gahlawat and Nicolas Nisse.