Given a c-colored graph G, a vertex of G is happy if it has the same color as all its neighbors. The notion of happy vertices was introduced by Zhang and Li to compute the homophily of a graph. Eto, et al. introduced the Maker-Maker version of the Happy vertex game, where two players compete to claim more happy vertices than their opponent. We introduce here the Maker-Breaker happy vertex game: two players, Maker and Breaker, alternately color the vertices of a graph with their respective colors. Maker aims to maximize the number of happy vertices at the end, while Breaker aims to prevent her. This game is also a scoring version of the Maker-Breaker Domination game introduced by Duchene, et al. as a happy vertex corresponds exactly to a vertex that is not dominated in the domination game. Therefore, this game is a very natural game on graphs and can be studied within the scope of scoring positional games. We initiate here the complexity study of this game, by proving that computing its score is PSPACE-complete on trees, NP-hard on caterpillars, and polynomial on subdivided stars. Finally, we provide the exact value of the score on graphs of maximum degree 2, and we provide an FPT-algorithm to compute the score on graphs of bounded neighborhood diversity. An important contribution of the paper is that, to achieve our hardness results, we introduce a new type of incidence graph called the literal-clause incidence graph for 2-SAT formulas. We prove that QMAX 2-SAT remains PSPACE-complete even if this graph is acyclic, and that MAX 2-SAT remains NP-complete, even if this graph is acyclic and has maximum degree 2, i.e. is a union of paths. We demonstrate the importance of this contribution by proving that Incidence, the scoring positional game played on a graph is also PSPACE-complete when restricted to forests.

翻译：给定一个c-着色图G，若G中某顶点与其所有邻接顶点颜色相同，则该顶点是快乐的。快乐顶点的概念由张和李引入，用于计算图的同质性。Eto等人提出了快乐顶点博弈的Maker-Maker版本，其中两名玩家竞争占据比对手更多的快乐顶点。本文引入Maker-Breaker快乐顶点博弈：两名玩家Maker和Breaker轮流用各自颜色为图的顶点着色。Maker的目标是最大化最终快乐顶点的数量，而Breaker的目标是阻止她实现这一目标。该博弈也是Duchene等人提出的Maker-Breaker支配博弈的计分版本，因为快乐顶点恰好对应支配博弈中未被支配的顶点。因此，这是一种非常自然的图博弈，可在计分位置博弈的框架下进行研究。我们首次对该博弈进行复杂性分析，证明了在树上计算其得分是PSPACE完全的，在毛虫图上是NP难的，在细分星图上是多项式可解的。最后，我们给出了最大度为2的图上得分的精确值，并提出了在邻域多样性有界的图上计算得分的FPT算法。本文的一个重要贡献是：为实现硬度结果，我们为2-SAT公式引入了一种新型关联图——文字-子句关联图。我们证明了即使该图是无环的，QMAX 2-SAT仍然是PSPACE完全的；即使该图是无环的且最大度为2（即路径的并集），MAX 2-SAT仍然是NP完全的。我们通过证明限制在森林上的关联图计分位置博弈Incidence也是PSPACE完全的，从而验证了这一贡献的重要性。