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

TL;DR

The paper analyzes Arc-Kayles and its non-disconnecting variant within impartial game theory, focusing on the complexity of subtraction games on graphs. It proves that for any finite subtraction set $S$ with $1 \notin S$, the connected subtraction game $\mathrm{CSG}(S)$ is PSPACE-complete even on bipartite graphs of even girth, and shows $\mathrm{CSG}(\{k\})$ is PSPACE-complete on split graphs for $k \ge 2$, along with GI-hardness for symmetry-based second-player strategies in Arc-Kayles. It then provides tractable results for Non-Disconnecting Arc-Kayles, including a quadratic kernel parameterized by the feedback edge number $f$ and polynomial-time algorithms on clique trees and certain threshold graphs, with a complexity gap highlighted between threshold and split graphs. The work advances understanding of when subtraction games on graphs remain intractable and identifies concrete, scalable algorithms and reductions, while outlining open questions on cases with $1 \in S$ and broader graph classes.

Abstract

Arc-Kayles is a game where two players alternate removing two adjacent vertices until no move is left, the winner being the player who played the last move. 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 give a quadratic kernel for Non-Disconnecting Arc-Kayles when parameterized by the feedback edge number, as well as polynomial-time algorithms for clique trees and a subclass 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.

Figures (7)

• Figure 1: An illustration of the reduction from Node-Kayles to $\mathop{\mathrm{CSG}}\nolimits(S)$ with $S = \{2,3\}$.
• Figure 2: The edge gadget $X_{uv}$ of the reduction of \ref{['thm-nonDisconnectingNodeKayles']}
• Figure 3: An illustration of the reduction from Avoid True to $\mathop{\mathrm{CSG}}\nolimits(k)$ with $k=2$. The clauses are $C_1=(x_1 \land x_2)$, $C_2=(x_2 \land x_4)$ and $C_3=(x_1 \land x_3)$. The $v_i$'s form a clique, and the other vertices an independent set.
• Figure 4: Example of graph having a pendant edge $(uv)$ such that $o(G) = \mathcal{N}$ and $o(G \setminus \{u,v\}) = \mathcal{N}$.
• Figure 5: An example of application of Reduction rule \ref{['red: replacing trees']}. Here, we have $o(T_u) = \mathcal{P}$ and $o(T_u-\{u\}) = \mathcal{N}$.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 4
• proof
• Definition 5