Independence Complexes and Edge Covering Complexes via Alexander Duality

The combinatorial Alexander dual of the independence complex Ind(G) and that of the edge covering complex EC(G) are shown to have isomorphic homology groups for each non-null graph G. This yields isomorphisms of homology groups of Ind(G) and EC(G) with homology dimensions being appropriately shifted and restricted. The results exhibits the complementary nature of homology groups of Ind(G) and EC(G) which had been proved by Ehrenborg-Hetyei [10], Engström [11], and Marietti-Testa [16] for forests at homotopy level.