Computational Complexity of Arc-Kayles and Non-Disconnecting Arc-Kayles Games on Graphs

Core Concepts

Arc-Kayles and Non-Disconnecting Arc-Kayles, two vertex deletion games on graphs, have their computational complexity studied. While Arc-Kayles remains open, Non-Disconnecting Arc-Kayles is shown to be polynomial-time solvable on certain structured graph classes.

Abstract

The paper studies the computational complexity of two vertex deletion games on graphs: Arc-Kayles and Non-Disconnecting Arc-Kayles.
Arc-Kayles is a two-player game where players alternate removing two adjacent vertices from a graph until no move is left. Its complexity has remained open since its introduction in 1978. The authors show that deciding whether a symmetry strategy can be applied for Arc-Kayles is as hard as the Graph Isomorphism problem, even on bipartite graphs.
Non-Disconnecting Arc-Kayles is a variant of Arc-Kayles where the players cannot disconnect the graph while removing vertices. This game is a subtraction game, a class of vertex deletion games with additional constraints. The authors prove that most subtraction games are PSPACE-complete, even on bipartite graphs of any given even girth. They also show that Non-Disconnecting Arc-Kayles is PSPACE-complete on split graphs.
On the positive side, the authors provide polynomial-time algorithms for solving Non-Disconnecting Arc-Kayles on several structured graph classes: unicyclic graphs, clique trees, and certain subclasses of threshold graphs. These results contrast with the open complexity of Arc-Kayles on even subdivided stars with three paths.

Stats

Arc-Kayles was introduced in 1978.
The complexity of Arc-Kayles remains open.
Non-Disconnecting Arc-Kayles is the subtraction game CSG({2}).
Non-Disconnecting Arc-Kayles is known to be polynomial-time solvable on trees, wheels, and grids of height at most 3.

Quotes

"Arc-Kayles is one of the many games introduced by Schaefer in his seminal paper on the computational complexity of games [20]. It is a two-player, information-perfect, finite vertex deletion game, in which the players alternate removing two adjacent vertices and all their incident edges from a graph."
"While Schaefer proved the PSPACE-completeness of many games, the complexity of Arc-Kayles surprisingly remains open."

Key Insights Distilled From

Complexity and algorithms for Arc-Kayles and Non-Disconnecting Arc-Kayles

by Kyle Burke,A... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10390.pdf

Deeper Inquiries

How can the complexity gap between threshold graphs and split graphs for Non-Disconnecting Arc-Kayles be further explored

To further explore the complexity gap between threshold graphs and split graphs for Non-Disconnecting Arc-Kayles, one could delve into the specific characteristics of threshold graphs that make them easier to solve compared to split graphs. This exploration could involve analyzing the structural properties of threshold graphs, such as their unique ordering of vertices and the presence of twin-free cliques. By investigating how these properties impact the gameplay and strategy in Non-Disconnecting Arc-Kayles, researchers can gain insights into why threshold graphs exhibit a different computational complexity profile compared to split graphs. Additionally, conducting a comparative analysis of gameplay scenarios in threshold and split graphs, focusing on critical moves and winning strategies, could shed light on the underlying reasons for the complexity gap.

Are there other structured graph classes where Non-Disconnecting Arc-Kayles can be solved efficiently

There are several other structured graph classes where Non-Disconnecting Arc-Kayles can be solved efficiently. One such class is cographs, which are graphs that can be constructed by repeatedly taking disjoint unions, joins, or complements of smaller cographs. Since cographs have a recursive structure, similar to threshold graphs, analyzing the gameplay dynamics of Non-Disconnecting Arc-Kayles on cographs could reveal patterns and strategies that lead to efficient solutions. Additionally, investigating the game on other tree-like graph classes, such as cacti or caterpillars, could provide further insights into the computational complexity of Non-Disconnecting Arc-Kayles on different graph structures.

What are the implications of the GI-hardness result for determining winning strategies in Arc-Kayles

The GI-hardness result for determining winning strategies in Arc-Kayles has significant implications for the game's complexity and solvability. The equivalence between verifying the existence of an involutive automorphism in a graph and the Graph Isomorphism problem implies that finding winning strategies in Arc-Kayles, particularly through symmetry strategies, is a computationally challenging task. This result suggests that determining optimal gameplay and predicting outcomes in Arc-Kayles, especially in scenarios involving symmetry considerations, may require sophisticated algorithms and computational resources. The GI-hardness underscores the intricate nature of strategy determination in Arc-Kayles and highlights the complexity involved in analyzing combinatorial games with symmetry properties.