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 alpha, beta the cost c(G) of a graph G is defined by c(G) = alpha vertical bar V(G)vertical bar + beta vertical bar E(G)vertical bar. We estimate the minimum cost of a (K-q; k)-vertex stable graph, i.e. a graph which contains a clique K-q after removing any k of its vertices. (C) 2013 Elsevier B.V. All rights reserved.