A generalization of an independent set with application to (Kq;k)-stable graphs

We introduce a natural generalization of an independent set of a graph and give a sharp lower bound on its size. The bound generalizes the widely known Caro and Wei result on the independence number of a graph. We use this result in the following problem. Given non-negative real numbers α,β the cost c(G) of a graph G is defined by c(G)=α|V(G)|+β|E(G)|. We estimate the minimum cost of a (Kq;k)-vertex stable graph, i.e. a graph which contains a clique Kq after removing any k of its vertices.