Recontamination Helps a Lot to Hunt a Rabbit

Thomas Dissaux, Foivos Fioravantes, Harmender Gahlawat, Nicolas Nisse

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

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 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 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 mh(G) of a graph G be the minimum integer k such that the Hunter player has a monotone winning strategy. We show that pw(G) ≤ mh(G) ≤ pw(G) + 1 for any graph G with pathwidth pw(G), which implies 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 ≥ 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.

Original languageEnglish
Title of host publication48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023
EditorsJerome Leroux, Sylvain Lombardy, David Peleg
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772921
DOIs
StatePublished - 1 Aug 2023
Event48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023 - Bordeaux, France
Duration: 28 Aug 20231 Sep 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume272
ISSN (Print)1868-8969

Conference

Conference48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023
Country/TerritoryFrance
CityBordeaux
Period28/08/231/09/23

Keywords

  • Graph Searching
  • Hunter and Rabbit
  • Monotonicity

ASJC Scopus subject areas

  • Software

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