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 $(K_{m,n};1)$-vertex stable graphs with minimum size. Namely, we prove that for m ≥ 2 and n ≥ m+2, $Q(K_{m,n};1) = mn+m+n$ and $K_{m,n}*K₁$ as well as $K_{m+1,n+1} - e$ are the only $(K_{m,n};1)$-vertex stable graphs with minimum size, confirming the conjecture of Dudek and Zwonek