  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 $P_n \times P_3 \times P_2$.
