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Complexity of Firefighting on Graphs

Authors Julius Althoetmar , Jamico Schade , Torben Schürenberg

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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Complexity
  • Cops and Robbers
  • Pursuit-Evasion

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Abstract

We consider a pursuit-evasion game that describes the process of extinguishing a fire burning on the nodes of an undirected graph. We denote the minimum number of firefighters required by ffn(G) and provide almost sharp bounds to this graph parameter for complete binary trees. We show that deciding whether ffn(G) ≤ m for given G and m is NP-hard. Furthermore, we show that shortest strategies can have superpolynomial length, leaving open whether the problem is in NP. We provide a construction that allows for transferring these results to a well-established Cops and Robbers variant called the "Hunter and Rabbit game".

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Julius Althoetmar, Jamico Schade, and Torben Schürenberg. Complexity of Firefighting on Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.4

Author Details

Julius Althoetmar
  • Technical University of Munich, Germany
Jamico Schade
  • Technical University of Munich, Germany
Torben Schürenberg
  • University of Bremen, Germany

Funding

  • Schürenberg, Torben: Deutsche Forschungsgemeinschaft (DFG) - Project number 281474342.

Acknowledgements

We would like to thank the participants of FRICO 2024, in particular Alexander Armbruster, and our colleagues for helpful discussions of the topic. Furthermore, we thank the reviewers for their insightful feedback.

References

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