# On minimum $({K}\_{q},k)$ stable graphs

Abstract : A graph G is a (Kq,k) stable graph (q ≥ 3) if it contains a Kq after deleting any subset of k vertices (k ≥ 0). Andrzej Żak in the paper On (Kq;k)-stable graphs, (/10.1002/jgt.21705) has proved a conjecture of Dudek, Szymański and Zwonek stating that for sufficiently large k the number of edges of a minimum (Kq,k) stable graph is (2q −3)(k+1) and that such a graph is isomorphic to sK2q −2+tK2q −3 where s and t are integers such that s(q −1)+t(q −2) −1 = k. We have proved (Fouquet et al. On (Kq,k) stable graphs with small k, Elektron. J. Combin. 19 (2012) #P50) that for q ≥ 5 and k ≤ q/2+1 the graph Kq+k is the unique minimum (Kq,k) stable graph. In the present paper we are interested in the (Kq, κ(q)) stable graphs of minimum size where κ(q) is the maximum value for which for every nonnegative integer k < κ(q) the only (Kq,k) stable graph of minimum size is Kq+k and by determining the exact value of κ(q).
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Jean-Luc Fouquet, Henri Thuillier, Jean-Marie Vanherpe, Adam Pawel Wojda. On minimum $({K}\_{q},k)$ stable graphs. Discussiones Mathematicae Graph Theory, University of Zielona Góra, 2013, 33 (1), pp.101-115. ⟨10.7151/dmgt.1656⟩. ⟨hal-00934301⟩

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