On (Kq;k)-Stable Graphs

A graph G is called (H; k)-vertex stable if G contains a subgraph isomorphic to H even after removing any k of its vertices. By stab(H; k) we denote the minimum size among the sizes of all (H; k)-vertex stable graphs. Given an integer q ≥ 2, we prove that, apart of some small values of k, stab(Kq; k) = (2q−3)(k+1). This confirms in the affirmative the conjecture of Dudek et al. [(H, k) stable graphs with minimum size, Discuss. Math. Graph Theory 28(1) (2008) 137–149]. Furthermore, we characterize the extremal graphs.