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:我们开始研究Maker-Breaker游戏在通用图形的边缘图形组中玩的算法复杂性。 我们主要考虑四大游戏中的三大游戏:连接游戏、完美的匹配游戏和$H$游戏。 如果Maker-Breaker游戏首先声称横贯树的边缘,第二个是完美的匹配,第三个是固定图形$H$的复制件。 我们证明决定谁赢得完美匹配游戏和$H$的游戏是PASCE, 即使后者是小直径图中的美元。 我们主要考虑找到最小的图表$H$的游戏。 因此, $H游戏是PASPCE游戏的完整。 我们还证明, 在正反面, 连接游戏和arrboria- game $H$的游戏是多盘时间游戏, 然后我们给$HCE($$) 提供了一些正面的结果, 首先给“Brebleer” 以最小直径的直径图, 4K 游戏是双平面的游戏。