PSPACE-completeness of the unrooted variants

Establish that the Unrooted Partizan Edge Geography, Unrooted Distinct Partizan Edge Geography, and Unrooted Trail Trap problems are PSPACE-complete, where the players choose their starting vertices before play.

Background

The rooted problems studied in the paper specify the initial positions of the blue and red tokens. The unrooted variants instead allow the players to choose their starting vertices before the game. The paper states a conjecture that both unrooted variants considered there are PSPACE-complete and relates them to Unrooted Partizan Edge Geography, also called Trail Trap, for which NP-hardness is known and PSPACE-completeness had previously been conjectured. The authors further suggest that resolving any one of these three closely related unrooted problems may provide sufficient insight to resolve the others.

References

We conjecture that the Unrooted variants, where the players can choose the starting vertices before the game, are also PSPACE-complete. A variant of Unrooted Partizan Edge Geography called Trail Trap is known to be NP-hard to test if the second player wins and the authors conjectured that the problem is PSPACE-completeConjecture 23.

— The Complexity of Undirected Partizan Edge Geography  (2609.18330 - Okada, 16 Sep 2026) in Section Conclusions