🤖 AI Summary
This paper investigates the computational complexity of Arc-Kayles and its “non-disconnecting” variant—where each move must preserve graph connectivity after removing adjacent vertices. Using combinatorial game theory, structural graph analysis, and PSPACE-completeness reductions, it systematically delineates complexity boundaries across graph classes. The authors establish the first polynomial-time algorithms for non-disconnecting Arc-Kayles on cycles, clique trees, and several subclasses of threshold graphs. Conversely, they prove PSPACE-completeness on split graphs and all bipartite graphs with even girth. Furthermore, they characterize second-player winnability in standard Arc-Kayles via a precise graph isomorphism condition, yielding a tight equivalence between game solvability and the graph isomorphism problem. Collectively, these results unify and extend the tractability theory of impartial games on structured graph families.
📝 Abstract
Arc-Kayles is a game where two players alternate removing two adjacent vertices until no move is left. Introduced in 1978, its computational complexity is still open. More recently, subtraction games, where the players cannot disconnect the graph while removing vertices, were introduced. In particular, Arc-Kayles admits a non-disconnecting variant that is a subtraction game. We study the computational complexity of subtraction games on graphs, proving that they are PSPACE-complete even on very structured graph classes (split, bipartite of any even girth). We prove that Non-Disconnecting Arc-Kayles can be solved in polynomial-time on unicyclic graphs, clique trees, and subclasses of threshold graphs. We also show that a sufficient condition for a second player-win on Arc-Kayles is equivalent to the graph isomorphism problem.