On vertex stability with regard to complete bipartite subgraphs

A graph G is called (H; k)-vertex stable if G contains a subgraph isomorphic to H ever after removing any of its k vertices. Q(H; k) denotes the minimum size among the sizes of all (H; k)-vertex stable graphs. In this paper we complete the characterization of (Km;n; 1)vertex stable graphs with minimum size. Namely, we prove that for m 2 and n m + 2, Q(Km;n; 1) = mn+m+n and Km;n K1 as well as Km+1;n+1 e are the only (Km;n; 1)-vertex stable graphs with minimum size, conrming the conjecture of Dudek and Zwonek.