We initiate the study of the algorithmic complexity of Maker-Breaker games played on the edge sets of general graphs. We mainly consider the perfect matching game and the $H$-game. Maker wins if she claims the edges of a perfect matching in the first, and a copy of a fixed graph $H$ in the second. We prove that deciding who wins the perfect matching game and the $H$-game is PSPACE-complete, even for the latter in small-diameter graphs if $H$ is a tree. Toward finding the smallest graph $H$ for which the $H$-game is PSPACE-complete, we also prove that such an $H$ of order 51 and size 57 exists. We then give several positive results for the $H$-game. As the $H$-game is already PSPACE-complete when $H$ is a tree, we mainly consider the case where $H$ belongs to a subclass of trees. In particular, we design two linear-time algorithms, both based on structural characterizations, to decide the winners of the $P_4$-game in general graphs and the $K_{1,\ell}$-game in trees. Then, we prove that the $K_{1,\ell}$-game in any graph, and the $H$-game in trees are both FPT parameterized by the length of the game, notably adding to the short list of games with this property, which is of independent interest. Another natural direction to take is to consider the $H$-game when $H$ is a cycle. While we were unable to resolve this case, we prove that the related arboricity-$k$ game is polynomial-time solvable. In particular, when $k=2$, Maker wins this game if she claims the edges of any cycle.

翻译：我们开启了对一般图边集上Maker-Breaker游戏算法复杂性的研究。主要考虑完美匹配游戏和$H$-游戏。在第一种游戏中，若Maker能声明完美匹配的所有边则获胜；在第二种游戏中，若她能声明固定图$H$的一个副本则获胜。我们证明判定完美匹配游戏和$H$-游戏的获胜方是PSPACE完全的，即使对于后者，当$H$为树且在直径较小的图上也是如此。为寻找使$H$-游戏成为PSPACE完全的最小图$H$，我们还证明存在一个阶数为51、边数为57的此类$H$。随后给出关于$H$-游戏的若干正面结果。由于当$H$为树时$H$-游戏已是PSPACE完全的，我们主要考虑$H$属于树子类的情形。特别地，基于结构刻画，我们设计了两个线性时间算法，分别用于判定一般图上的$P_4$-游戏和树上的$K_{1,\ell}$-游戏的获胜方。接着证明：任意图上的$K_{1,\ell}$-游戏和树上的$H$-游戏均以游戏步长为参数具有固定参数易处理性（FPT），这显著增添了拥有该性质的少量游戏实例，具有独立研究价值。另一个自然方向是考虑$H$为圈时的$H$-游戏。尽管未能解决此情形，我们证明相关的树状度-$k$游戏可在多项式时间内求解。特别地，当$k=2$时，若Maker能声明任意圈的所有边，则她获胜。