We initiate the study of the algorithmic complexity of Maker-Breaker games played on edge sets of graphs for general graphs. We mainly consider three of the big four such games: the connectivity game, perfect matching game, and $H$-game. Maker wins if she claims the edges of a spanning tree in the first, a perfect matching in the second, and a copy of a fixed graph $H$ in the third. We prove that deciding who wins the perfect matching game and the $H$-game is PSPACE-complete, even for the latter in graphs of small diameter if $H$ is a tree. Seeking to find the smallest graph $H$ such that the $H$-game is PSPACE-complete, we also prove that there exists such an $H$ of order 51 and size 57. On the positive side, we show that the connectivity game and arboricity-$k$ game are polynomial-time solvable. We then give several positive results for the $H$-game, first giving a structural characterization for Breaker to win the $P_4$-game, which gives a linear-time algorithm for the $P_4$-game. We provide a structural characterization for Maker to win the $K_{1,\ell}$-game in trees, which implies a linear-time algorithm for the $K_{1,\ell}$-game in trees. Lastly, 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. We leave the complexity of the last of the big four games, the Hamiltonicity game, as an open question.

翻译：本文首次研究了一般图边集上Maker-Breaker游戏的算法复杂性。我们主要关注此类游戏中四大经典问题中的三个：连通性游戏、完美匹配游戏和$H$-游戏。Maker若在第一个游戏中获得一棵生成树的边、在第二个游戏中获得一个完美匹配、在第三个游戏中获得固定图$H$的一个副本即获胜。我们证明判定完美匹配游戏和$H$-游戏胜负的问题是PSPACE完全的——即使对于后者，当$H$为树且图直径较小时亦如此。为寻找使$H$-游戏成为PSPACE完全问题的最小图$H$，我们进一步证明存在一个阶数为51、边数为57的这样的图$H$。在积极方面，我们证明连通性游戏和树状性-$k$游戏可在多项式时间内求解。随后给出$H$-游戏的若干正向结果：首先给出Breaker赢得$P_4$-游戏的结构特征，由此提出$P_4$-游戏的线性时间算法；其次给出Maker在树中赢得$K_{1,\ell}$-游戏的结构特征，并导出树中$K_{1,\ell}$-游戏的线性时间算法。最后证明任意图中的$K_{1,\ell}$-游戏和树中的$H$-游戏均关于游戏时长参数为FPT。我们留下四大经典游戏中最后一个——哈密顿性游戏——的复杂性作为开放问题。