Dominance complexes, neighborhood complexes and combinatorial Alexander duals

We show that the dominance complex $\mathcal{D}(G)$ of a graph $G$ coincides with the combinatorial Alexander dual of the neighborhood complex $\mathcal{N}(\overline{G})$ of the complement of $G$. Using this, we obtain a relation between the chromatic number $\chi(G)$ of $G$ and the homology group of $\mathcal{D}(G)$. 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 $\mathcal{D}(P_n \times P_3)$ by determining the homotopy types of the independence complex of $P_n \times P_3 \times P_2$.