Geodesic
On digraphs with a~small number of arcs in a~minimal $1$-vertex extension
Prikladnaya Diskretnaya Matematika. Supplement, no. 5 (2012), pp. 86-88

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A graph $G^*$ nodes is vertex extension of graph $G$ with $n$ nodes if every graph obtained by removing any vertex from $G^*$ contains $G$. Vertex extension of graph $G$ with $n+1$ vertices is called minimal if among all vertex extensions of graph $G$ with $n+1$ vertices it has the minimum possible number of edges. We study digraphs, whose minimal vertex extensions have a specified number of additional arcs. A solution is given when the number of additional arcs is equal to one or two. 
@article{PDMA_2012_5_a43,
     author = {M. B. Abrosimov and O. V. Modenova},
     title = {On digraphs with a~small number of arcs in a~minimal $1$-vertex extension},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {86--88},
     publisher = {mathdoc},
     number = {5},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2012_5_a43/}
}
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M. B. Abrosimov; O. V. Modenova. On digraphs with a~small number of arcs in a~minimal $1$-vertex extension. Prikladnaya Diskretnaya Matematika. Supplement, no. 5 (2012), pp. 86-88. http://geodesic.mathdoc.fr/item/PDMA_2012_5_a43/