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{\bf Sylwia Cichacz, Agnieszka G\"{o}rlich,  Magorzata Zwonek and Andrzej \.{Z}ak}
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{\bf On $(C_n;k)$ Stable Graphs}
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A graph $G$ is called $(H;k)$-{\it vertex stable} if $G$ contains a subgraph
isomorphic to $H$ ever after removing any $k$ of its  vertices; stab$(H;k)$
denotes the minimum size among the sizes of all $(H;k)$-vertex stable
graphs. In this paper we deal with $(C_{n};k)$-vertex stable graphs with minimum
size. For each $n$ we prove that stab$(C_{n};1)$ is one of only two possible values and
we give the exact value for infinitely many $n$'s. Furthermore we establish
an upper and lower bound for stab$(C_{n};k)$ for $k\geq 2$.

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