Further results on the Hunters and Rabbit game through monotonicity

TL;DR

This paper introduces a monotonic variant of the Hunters and Rabbit game, establishes its relation to graph pathwidth, analyzes its computational complexity, and provides polynomial algorithms for specific graph classes.

Contribution

It defines the monotone hunter number, relates it to pathwidth, and shows polynomial-time computability in certain graph classes, highlighting differences from the standard hunter number.

Findings

Monotone hunter number is closely related to pathwidth, differing by at most 1.

Computing the monotone hunter number is NP-hard in general graphs.

Polynomial algorithms exist for split graphs, interval graphs, cographs, and trees.

Abstract

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 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 $h(G)$ 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 $h(G)$ remains open in general graphs and even in trees. To progress further, we propose a notion of monotonicity 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. Let the monotone hunter number be denoted by $mh(G)$. We show that $pw(G) \leq mh(G) \leq pw(G)+1$ for any graph $G$ with pathwidth $pw(G)$, implying that computing $mh(G)$, or even approximating $mh(G)$ up to an additive constant, is NP-hard. Then, we show that $mh(G)$ 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 $h$ and $mh$, i.e., that monotonicity does not help. In particular, we show that, for every $k\geq 3$, there exists a tree $T$ with $h(T)=2$ and $mh(T)=k$. We conclude by proving that computing $h$ (resp., $mh$) is FPT parameterised by the minimum size of a vertex cover.