Complexity of deciding whether the hunting number is at most a fixed constant

Determine, for each fixed integer k greater than or equal to 2, the computational complexity of deciding whether an input graph G satisfies h(G) <= k, including the restricted case in which G is a tree.

Background

The paper establishes NP-hardness for computing the hunting number on bipartite simple graphs and related restricted instances, but it does not resolve the complexity of the threshold decision problem for any fixed number of hunters k >= 2. The authors specifically note that the question remains unresolved even for trees, despite the fact that deciding whether one hunter can win is polynomial-time solvable.

References

We know that deciding whether one hunter can win is polynomial, but the complexity of deciding whether the hunting number is less than some given constant $k$ is unknown for $k\ge 2$ even in the case of trees.

— Hunting a rabbit is hard  (2502.15982 - Ben-Ameur et al., 21 Feb 2025) in Section 6, Final remarks