Geodesic
Characterization of graphs with a~small number of additional arcs in a~minimal $1$-vertex extension
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 2, pp. 3-9

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A graph $G^*$ is a $k$-vertex extension of a graph $G$ if every graph obtained from $G^*$ by removing any $k$ vertices contains $G$. $k$-vertex extension of a graph $G$ with $n+k$ vertices is called minimal if among all $k$-vertex extensions of $G$ with $n+k$ vertices it has the minimal possible number of arcs. We study directed graphs, whose minimal vertex $1$-extensions have a specific number of additional arcs. A solution is given when the number of additional arcs equals one or two. 
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     title = {Characterization of graphs with a~small number of additional arcs in a~minimal $1$-vertex extension},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
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M. B. Abrosimov; O. V. Modenova. Characterization of graphs with a~small number of additional arcs in a~minimal $1$-vertex extension. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 2, pp. 3-9. http://geodesic.mathdoc.fr/item/ISU_2013_13_2_a0/