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The combinatorial Alexander dual of the independence complex $\mathrm{Ind}(G)$ and that of the edge covering complex $\mathrm{EC}(G)$ are shown to have isomorphic homology groups for each non-null graph $G$. This yields isomorphisms of homology groups of $\mathrm{Ind}(G)$ and $\mathrm{EC}(G)$ with homology dimensions being appropriately shifted and restricted. The results exhibits the complementary nature of homology groups of $\mathrm{Ind}(G)$ and $\mathrm{EC}(G)$ which had been proved by Ehrenborg-Hetyei, Engstrom, and Marietti-Testa for forests at homotopy level.

Independence Complexes and Edge Covering Complexes via Alexander Duality