Geodesic
On the number of additional edges of a~minimal vertex 1-extension of a~starlike tree
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 12 (2012) no. 2, pp. 103-113.

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For a given graph $G$ with $n$ nodes, we say that graph $G^*$ is its 1-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 1-extension of the graph $G$ if $G^*$ has $n+1$ nodes and there is no 1-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 1-extension of a starlike tree and present trees on which these bounds are achieved. 
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M. B. Abrosimov. On the number of additional edges of a~minimal vertex 1-extension of a~starlike tree. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 12 (2012) no. 2, pp. 103-113. http://geodesic.mathdoc.fr/item/ISU_2012_12_2_a14/

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