🤖 AI Summary
This study investigates the computational complexity of Edge Geography under various graph width parameters, including pathwidth and tree-partition width. Leveraging parameterized complexity theory, graph decomposition techniques, and XNLP-hardness reductions, the work establishes that Edge Geography is XNLP-hard when parameterized by pathwidth and belongs to the XP class with respect to tree-partition width. Furthermore, it presents a fixed-parameter tractable algorithm for the directed variant when parameterized jointly by treewidth and maximum degree. These results resolve an open problem posed by Bodlaender over three decades ago and provide a complete characterization of the precise parameterized complexity landscape of Edge Geography across mainstream width measures.
📝 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.