Misère Partizan Arc Kayles is PSPACE-complete, even on Planar Graphs

TL;DR

The paper proves that Misère Partizan Arc Kayles on planar graphs is $PSPACE$-complete. It achieves this via a reduction chain from PositiveCNF through Bounded Two-Player Constraint Logic, with grid-embedded gadget construction on square and triangular lattices. The authors introduce and analyze three PSPACE-hard variants of B2CL (BBB2CL, NPB2CL, MPB2CL) and describe a full set of basis gadgets (Variable, And, Or, Split, Choice, Goal) to realize the reduction. This work extends constraint-logic techniques into misère partizan settings, providing a framework for future hardness results on planar graphs and related games such as Arc Kayles and Domineering.

Abstract

We show that Misère Partizan Arc Kayles is PSPACE-complete on planar graphs via a reduction from Bounded Two-Player Constraint Logic. Furthermore, we show how to embed our gadgets onto the square and triangular grids. In order to clearly explain these results, we get into the details of Bounded Two-Player Constraint Logic and find three PSPACE-complete variants of that as well.

Figures (23)

• Figure 1: B2CL Gadgets.
• Figure 2: Translated from Figure 6.1 of DBLP:books/daglib/0023750, a B2CL position equivalent to the PosCNF game on $(w \vee x \vee y) \wedge (w \vee z) \wedge (x \vee z)$. For space reasons (as in the source figure) it does not contain all simplifications to only the five basis vertices and does not employ the crossover gadgets. We do, however, include the extra $k$ hidden red plays described.
• Figure 3: Modification of the Blue goal from the reduction to reach NPB2CL. On the left side is the goal part of the original B2CL and BBB2CL reductions. On the right side is the modification using additional Or gadgets that gives Blue $2k$ additional moves.
• Figure 4: New basis vertices needed for the reduction: Blue-to-Red (left) and Red-Or (right).
• Figure 5: Modification of the Blue goal from the reduction to reach MPB2CL. On the left side is the goal part of the original B2CL and BBB2CL reductions. On the right side is the modification using additional gadgets that gives Red $2k$ moves if Blue would have met the goal originally.

Theorems & Definitions (17)

• Definition 2.1: Builder-Blocker Bounded Two-Player Constraint Logic
• Definition 2.2: Normal-Play Bounded Two-Player Constraint Logic
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Lemma 2.5
• proof
• Corollary 2.6
• proof