The Domination game is an impartial game on graphs, introduced in 2010, and proved PSPACE-complete in the normal variant in 2026. In this game, Alice and Bob alternately select playable vertices, where a vertex is playable if it dominates at least one vertex not dominated by the vertices selected before in the game. The game ends when the selected vertices form a dominating set. In the normal variant, the player unable to move loses. In contrast to the impartial game, the partizan game has the vertices already colored with $A$, $B$, or $C$, in such a way that Alice (resp. Bob) can only select vertices colored with $A$ (resp. $B$) or $C$. The partizan game was proved PSPACE-hard in 2026. In this paper, we determine the winner of the Normal Partizan Domination game in graphs whose components are complete split graphs, including star forests, for any initial coloring of its vertices. We also obtain partial results for complete bipartite graphs.

翻译：支配博弈是图上的一个无偏博弈，于2010年提出，其标准变体在2026年被证明为PSPACE完全问题。在该博弈中，Alice和Bob轮流选择可玩顶点，其中若一个顶点至少支配一个未被游戏中先前选择的顶点所支配的顶点，则该顶点为可玩顶点。当所选顶点构成一个支配集时博弈结束。在标准变体中，无法移动的玩家输掉比赛。与无偏博弈不同，偏袒博弈的顶点已预先标记为$A$、$B$或$C$，使得Alice（相应地，Bob）只能选择标记为$A$（相应地，$B$）或$C$的顶点。偏袒博弈于2026年被证明为PSPACE难问题。在本文中，我们确定了分量是完全分裂图（包括星形森林）的图中，对于任意初始顶点着色，标准偏袒支配博弈的胜者。我们还得出了完全二分图的部分结果。