  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.
