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Edge Geography is XNLP-hard for Pathwidth and in XP for Tree-Partition Width

Authors Thobias Kvalvik Høivik , Erlend Raa Vågset

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LIPIcs.ESA.2026.114.pdf
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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Geography games
  • graph games
  • pathwidth
  • parameterized complexity
  • XNLP-hardness
  • PSPACE-complete games
  • tree-partition width

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Abstract

Directed Edge Geography and Undirected Edge Geography are classical PSPACE-complete two-player graph games in which players alternately make moves along edges, deleting each one after use; the first player unable to move loses. We prove that both problems are XNLP-hard when parameterized by pathwidth, addressing a question raised by Bodlaender over 30 years ago. On the positive side, we observe that Directed Edge Geography is fixed-parameter tractable when parameterized by treewidth and maximum degree. We also prove that both problems are in XP on simple graphs when parameterized by tree-partition width. These results develop modern lower-bound and decomposition-based algorithmic methods for width-based questions in PSPACE-complete graph games.

Cite As Get BibTex

Thobias Kvalvik Høivik and Erlend Raa Vågset. Edge Geography is XNLP-hard for Pathwidth and in XP for Tree-Partition Width. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 114:1-114:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.114

Author Details

Thobias Kvalvik Høivik
  • Department of Computer Science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences, Førde, Norway
Erlend Raa Vågset
  • Department of Computer Science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences, Førde, Norway

Funding

Thobias Kvalvik Høivik acknowledges support from StudentFORSK at Western Norway University of Applied Sciences. Erlend Raa Vågset acknowledges support from the HVL Robotics research group at Western Norway University of Applied Sciences.

Acknowledgements

We thank the anonymous reviewers for their careful reading and for their thoughtful and constructive feedback, which helped us improve the exposition and clarify several aspects of the paper.

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References

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