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