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arXiv · 2607.03189

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

Thobias Kvalvik Høivik·Erlend Raa Vågset
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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.

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Thobias Kvalvik Høivik, Erlend Raa Vågset. 2026-07-03. Edge Geography is XNLP-hard for Pathwidth and in XP for Tree-Partition Width. https://doi.org/10.4230/lipics.esa.2026.114

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