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/

[1] R. Diestel, Graph Theory, second ed. (Springer-Verlag, 2000).

[2] A. Dudek, A. Szymaski and M. Zwonek, (H,k) stable graphs with minimum size, Discuss. Math. Graph Theory 28 (2008) 137-149, doi: 10.7151/dmgt.1397.

[3] A. Dudek and M. Zwonek, (H,k) stable bipartite graphs with minimum size, Discuss. Math. Graph Theory 29 (2009) 573-581, doi: 10.7151/dmgt.1465.