Geodesic
On vertex stability with regard to complete bipartite subgraphs
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 663-669

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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.
Keywords: vertex stable, bipartite graph, minimal size 
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Dudek, Aneta; Żak, Andrzej. On vertex stability with regard to complete bipartite subgraphs. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 4, pp. 663-669. http://geodesic.mathdoc.fr/item/DMGT_2010_30_4_a10/