Domination numbers and homology

Let I(G) denote the independence complex of a graph G=(V,E). Some relations between domination numbers of G and the homology of I(G) are given. As a consequence the following Hall-type conjecture of Aharoni is proved: Let γs∗(G) denote the fractional star-domination number of G and let V=⋃i=1mVi be a partition of V into m classes. (G[⋃i∈IVi])>|I|−1 for all I⊂{1,…,m} then G contains an independent set which intersects all m classes.