We show that the dominance complex D(G) of a graph G coincides with the combinatorial Alexander dual of the neighborhood complex N(G‾) of the complement of G. Using this, we obtain a relation between the chromatic number χ(G) of G and the homology group of D(G). We also obtain several known results related to dominance complexes from well-known facts of neighborhood complexes. After that, we suggest a new method for computing the homology groups of the dominance complexes, using independence complexes of simple graphs. We show that several known computations of homology groups of dominance complexes can be reduced to known computations of independence complexes. Finally, we determine the homology group of D(Pn×P3) by determining the homotopy types of the independence complex of Pn×P3×P2.

中文翻译：

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显性复合体、邻域复合体和组合 Alexander 对偶


我们表明，图 G 的显性复合体 D（G） 与 G 的补数的邻域复合体 N（G ̅） 的组合亚历山大对偶重合。利用这一点，我们获得了 G 的色数 χ（G） 和 D（G） 的同源群之间的关系。我们还从邻域复合体的已知事实中获得了几个与显性复合体相关的已知结果。之后，我们提出了一种使用简单图的独立复数计算显性复合体的同源群的新方法。我们表明，显性复合体的同源群的几种已知计算可以简化为独立的复合体的已知计算。最后，我们通过确定 Pn×P3×P2 独立复合物的同伦类型来确定 D（Pn×P3） 的同源群。