A dominating set of a graph is a subset D of the vertices such that every vertex not in D is adjacent to some vertex of D. In this paper, we introduce several variants of dominating sets in the square grid, periodic square grid and cylindric square grid by considering translation symmetry. We provide their exact enumerations in terms of domination polynomials. We also analyze the asymptotic behavior of the growth rates of their cardinality.

中文翻译：

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圆柱方格图中的支配集数

图的支配集是顶点的子集D，使得每个不在D中的顶点都与D 的某个顶点相邻。在本文中，我们通过考虑平移对称性，介绍了方格、周期方格和圆柱方格中支配集的几种变体。我们根据支配多项式提供了它们的精确枚举。我们还分析了它们的基数增长率的渐近行为。