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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. Mader, “Critically n-connected digraphs,” in: Graph Theory, Combinatorics, and Applications, Vol. 2 (1991), pp. 811–829.
J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms, and Applications (2000).
S. V. Savchenko, “On the number of noncritical vertices in strongly connected digraphs,” Mat. Zametki, 79, 743–755 (2006).
S. V. Savchenko, “On the number of non-critical vertices in strong tournaments of order N with minimum out-degree δ + and in-degree δ −,” Discrete Math., 310, 1177–1183 (2010).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 406, 2012, pp. 107–116.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1694-5