Let D be a strongly connected digraph on n ≥ 4 vertices. A vertex v of D is noncritical if the digraph D − v is strongly connected. It is proved that if the sum of degrees of any two adjacent vertices of D is at least n + 1, then there exists a noncritical vertex in D, and if the sum of degrees of any two adjacent vertices of D is at least n + 2, then there exist two noncritical vertices in D. A series of examples confirms that these bounds are tight. Bibliography: 4 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 406, 2012, pp. 107–116.
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Nenashev, G.V. On the Existence of Noncritical Vertices in Digraphs. J Math Sci 196, 791–796 (2014). https://doi.org/10.1007/s10958-014-1694-5
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DOI: https://doi.org/10.1007/s10958-014-1694-5