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 gamma(s)(*)(G) denote the fractional star-domination number of G and let V = boolean ORi=1m V-i be a partition of V into m classes. If gamma(s)* (G[boolean ORiis an element ofI Vi]) > \I\ - 1 for all I subset of {1,..., m} then G contains an independent set which intersects all m classes. (C) 2003 Elsevier Science (USA). All rights reserved.