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arXiv · 2610.18537

PSPACE-completeness of the Hunters and Rabbit game

Walid Ben-Ameur·Alessandro Maddaloni
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Abstract

In the Hunters and Rabbit game, hunters shoot at a set of vertices at each round, while an invisible rabbit must move along an edge. We prove that deciding whether a given number of hunters can guarantee capture is PSPACE-complete, even on simple bipartite graphs. The reduction uses intersection nonemptiness for inverse finite automata: one common word must put all automata in accepting states simultaneously. A clique acting as a safe haven and two families of guards force every winning strategy to simulate common transitions. Existing transformations remove the restriction on the rabbit's initial positions and yield PSPACE-completeness for the directed game with one hunter, even on oriented graphs with positive minimum outdegree. Adjusting the guard sizes also proves PSPACE-completeness of the firefighting game, in which the rabbit may remain at its current vertex.

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Walid Ben-Ameur, Alessandro Maddaloni. 2026-10-13. PSPACE-completeness of the Hunters and Rabbit game. https://arxiv.org/abs/2610.18537

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