Node Kayles is a well-known two-player impartial game on graphs: Given an undirected graph, each player alternately chooses a vertex not adjacent to previously chosen vertices, and a player who cannot choose a new vertex loses the game. The problem of deciding if the first player has a winning strategy in this game is known to be PSPACE-complete. There are a few studies on algorithmic aspects of this problem. In this paper, we consider the problem from the viewpoint of fixed-parameter tractability. We show that the problem is fixed-parameter tractable parameterized by the size of a minimum vertex cover or the modular-width of a given graph. Moreover, we give a polynomial kernelization with respect to neighborhood diversity.

中文翻译：

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关于节点 Kayles 的结构参数化

Node Kayles 是一个著名的图上的两人公平博弈：给定一个无向图，每个玩家交替选择一个与先前选择的顶点不相邻的顶点，一个不能选择新顶点的玩家输掉游戏。在这个游戏中决定第一个玩家是否有获胜策略的问题已知是 PSPACE-complete。有一些关于这个问题的算法方面的研究。在本文中，我们从固定参数易处理性的角度考虑问题。我们表明该问题是由最小顶点覆盖的大小或给定图的模宽度参数化的固定参数易处理问题。此外，我们给出了关于邻域多样性的多项式核化。