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Keywords:

  • complete graphs;
  • vertex stability

Abstract

A graph G is called inline image-vertex stable if G contains a subgraph isomorphic to H even after removing any k of its vertices. By stabinline image we denote the minimum size among the sizes of all inline image-vertex stable graphs. Given an integer inline image, we prove that, apart of some small values of k, stabinline image. 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.