Geodesic
On a~counterexample to a~minimal vertex $1$-extension of starlike trees
Prikladnaya Diskretnaya Matematika. Supplement, no. 5 (2012), pp. 83-84.

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For a given graph $G$ with $n$ nodes, we say that graph $G^*$ is its vertex extension if for each vertex $v$ of $G^*$ the subgraph $G^*-v$ contains graph $G$ up to isomorphism. A graph $G^*$ is a minimal vertex extension of the graph $G$ if $G^*$ has $n+1$ nodes and there is no vertex extension with $n+1$ nodes of $G$ having fewer edges than $G^*$. A tree is called starlike if it has exactly one node of degree greater than two. We give a lower and upper bounds of the edge number of a minimal vertex extension of a starlike tree and present trees for which these bounds are achieved. 
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M. B. Abrosimov; D. D. Komarov. On a~counterexample to a~minimal vertex $1$-extension of starlike trees. Prikladnaya Diskretnaya Matematika. Supplement, no. 5 (2012), pp. 83-84. http://geodesic.mathdoc.fr/item/PDMA_2012_5_a41/

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[2] Harary F., Khurum M., “One node fault tolerance for caterpillars and starlike trees”, Internet J. Comput. Math., 56 (1995), 135–143 | DOI | Zbl

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