Graphs whose every independent set has a common neighbour

A graph is said to have property I if every independent set of its vertices has a common neighbour. The following conjecture of P. Erdos and S. Fajtlowicz is proved: There exists a constant c>0 such that any triangle free graph with property I has a vertex of degree >=c|V(G)|. (This is true with c=13 and this value is best possible.) The complete characterization of the graphs having the above properties is given. In the last section we deal with graphs satisfying some weaker conditions.