We study 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：本文研究了在一般图边集上进行的Maker-Breaker游戏的算法复杂度。我们主要考察完美匹配游戏和$H$-游戏：在第一种游戏中，若Maker占据的边构成一个完美匹配则获胜；在第二种游戏中，若Maker占据的边包含一个固定图$H$的副本则获胜。我们证明了判定完美匹配游戏与$H$-游戏的获胜方均为PSPACE完全问题，即使对于后者在直径较小的图中（当$H$为树时）亦然。为寻找使$H$-游戏具有PSPACE完全性的最小图$H$，我们进一步证明存在一个阶数为51、边数为57的图$H$满足该性质。随后我们针对$H$-游戏给出若干正面结果。鉴于当$H$为树时$H$-游戏已是PSPACE完全的，我们主要研究$H$属于树之子类的情况。特别地，基于结构特征我们设计了两种线性时间算法，分别用于判定一般图中$P_4$-游戏的获胜方以及树中$K_{1,\ell}$-游戏的获胜方。接着我们证明，任意图中的$K_{1,\ell}$-游戏以及树中的$H$-游戏均是关于游戏长度的固定参数可解（FPT）问题，这为具有该性质的游戏短列表增添了新案例，具有独立研究价值。另一个自然的研究方向是考虑$H$为环时的$H$-游戏。虽然未能完全解决该情形，但我们证明了相关的树性-$k$游戏可在多项式时间内求解。特别地，当$k=2$时，若Maker占据的边包含任意环则获胜。