Mathematical Notes

, Volume 79, Issue 5–6, pp 687–696 | Cite as

On the number of noncritical vertices in strongly connected digraphs

  • S. V. Savchenko


By definition, a vertex w of a strongly connected (or, simply, strong) digraph D is noncritical if the subgraph D — w is also strongly connected. We prove that if the minimal out (or in) degree k of D is at least 2, then there are at least k noncritical vertices in D. In contrast to the case of undirected graphs, this bound cannot be sharpened, for a given k, even for digraphs of large order. Moreover, we show that if the valency of any vertex of a strong digraph of order n is at least 3/4n, then it contains at least two noncritical vertices. The proof makes use of the results of the theory of maximal proper strong subgraphs established by Mader and developed by the present author. We also construct a counterpart of this theory for biconnected (undirected) graphs.

Key words

digraph strongly connected digraph biconnected graph critical vertex maximal proper strong subdigraph Hamiltonian cycle 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • S. V. Savchenko
    • 1
  1. 1.L. D. Landau Institute for Theoretical PhysicsRussia

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