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Nemesis, an Escape Game in Graphs

Authors Pierre Bergé , Antoine Dailly, Yan Gerard

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LIPIcs.FUN.2026.7.pdf
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  • The full version including a variant of the game and the NP-completeness of the computation of a binary escape tree is hosted on ArXiv.
  • Full Version https://arxiv.org/abs/2601.13841

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Graphs
  • Evasion and Pursuit Games
  • PSPACE-completeness
  • Quantified SAT
  • Canadian Traveler Problem
  • Cat Herding Problem

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Abstract

We define a new escape game in graphs that we call Nemesis. The game is played on a graph having a subset of vertices labeled as exits and the goal of one of the two players, called the fugitive, is to reach one of these exit vertices. The second player, i.e. the fugitive adversary, is called the Nemesis. Her goal is to trap the fugitive in a connected component which does not contain any exit. At each round of the game, the fugitive moves from one vertex to an adjacent vertex. Then the Nemesis deletes one edge anywhere in the graph. The game ends when either the fugitive reached an exit or when he is in a connected component that does not contain any exit. In trees and graphs of maximum degree bounded by 3, Nemesis can be solved in linear time. For arbitrary graphs, we show that Nemesis is PSPACE-complete, and that it is NP-hard on planar multigraphs. We extend our results to the related Cat Herding problem, proving its PSPACE-completeness.

Cite As Get BibTex

Pierre Bergé, Antoine Dailly, and Yan Gerard. Nemesis, an Escape Game in Graphs. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 7:1-7:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.7

Author Details

Pierre Bergé
  • Université Grenoble Alpes, CNRS, Grenoble INP, LIG, 38000 Grenoble, France
Antoine Dailly
  • Université Clermont Auvergne, INRAE, UR TSCF, 63000 Clermont-Ferrand, France
Yan Gerard
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France

Funding

This work was supported by the project ANR SPAWN ANR-25-CE48-1750.

Acknowledgements

We thank Claire Hilaire and Gaëtan Berthe for the discussions and the different ideas they brought to this work.

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