The Normal Play of the Domination Game

In 2010, Bre\v{s}ar, Klav\v{z}ar and Rall introduced the optimization variant of the graph domination game and the game domination number, which was proved PSPACE-hard by Bre\v{s}ar et al. in 2016. In 2024, Leo Versteegen obtained the celebrated proof of the Conjecture $\frac{3}{5}$ on this variant of the domination game, proposed by Kinnersley, West and Zamani in 2013. In this paper, we investigate for the first time the normal play of the domination game, which we call Normal Domination Game, that is an impartial game where the last to play wins. We first prove that this game is PSPACE-complete even in graphs with diameter two. We also use the Sprague-Grundy theory to prove that Alice (the first player) wins in the path $P_n$ if and only if $n$ is not a multiple of $4$, and wins in the cycle $C_n$ if and only if $n=4k+3$ for some integer $k$. Moreover, we obtain a polynomial time algorithm to decide the winner for any disjoint union of paths and cycles in the Normal Domination Game and its natural partizan variant. Finally, we also prove that the Mis\`ere Domination Game (the last to play loses) is PSPACE-complete, as are the natural partizan variants of the normal game and the mis\`ere game.