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On math formula-Stable Graphs



A graph G is called math formula-vertex stable if G contains a subgraph isomorphic to H even after removing any k of its vertices. By stabmath formula we denote the minimum size among the sizes of all math formula-vertex stable graphs. Given an integer math formula, we prove that, apart of some small values of k, stabmath formula. This confirms in the affirmative the conjecture of Dudek et al. [Discuss Math Graph Theory 28(1) (2008), 137–149]. Furthermore, we characterize the extremal graphs.