Advertisement

Journal of Mathematical Sciences

, Volume 212, Issue 6, pp 654–665 | Cite as

The Tree of Cuts and Minimal k-Connected Graphs

  • D. V. Karpov
Article
  • 30 Downloads
A cut of a k-connected graph G is a k-element cutset of it, which contains at least one edge. The tree of cuts of a set , consisting of pairwise independent cuts of a k-connected graph, is defined in the following way. Its vertices are the cuts of the set and the parts of the decomposition of G induced by these cuts. A part A is adjacent to a cut S if and only if A contains all the vertices of S and exactly one end of each edge of S. It is proved that the defined graph is a tree, some properties of which are similar to the corresponding properties of the classic tree of blocks and cutpoints.

In the second part of the paper, the tree of cuts is used to study minimal k-connected graphs for k ≤ 5. Bibliography: 11 titles.

Keywords

Bipartite Graph Common Edge Distinct Edge Mutual Disposition Minimal Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. A. Dirac, “Minimally 2-connected graphs,” J. Reine Angew. Math., 268, 204–216 (1967).MathSciNetGoogle Scholar
  2. 2.
    M. D. Plummer, “On minimal blocks,” Trans. Amer. Math. Soc., 134, 85–94 (1968).MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    W. Tutte, Connectivity in Graphs, Univ. Toronto Press, Toronto (1966).MATHGoogle Scholar
  4. 4.
    W. Mader, “Ecken Vom Gard n in minimalen n-fach zusammenhangenden Graphen,” Arch. Math., 23, 219–224 (1972).MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    W. Mader, “On vertices of degree n in minimally n-connected graphs and digraphs,” in: Combinatorics, Paul Erd¨os is Eighty, Vol. 2, Budapest (1996), pp. 423–449.Google Scholar
  6. 6.
    W. Mader, “Zur Struktur minimal n-fach zusammenh¨angender Graphen,” Abh. Math. Sem. Univ. Hamburg, 49, 49–69 (1979).MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    F. Harary, Graph Theory, Addison-Wesley (1969).Google Scholar
  8. 8.
    D. V. Karpov and A. V. Pastor, “The structure of a decomposition of a triconnected graph,” Zap. Nauchn. Semin. POMI, 391, 90–148 (2011).Google Scholar
  9. 9.
    D. V. Karpov, “Cutsets in a k-connected graph,” Zap. Nauchn. Semin. POMI, 340, 33–60 (2006).MATHGoogle Scholar
  10. 10.
    D. V. Karpov, “The tree of decomposition of a biconnected graph,” Zap. Nauchn. Semin. POMI, 417, 86–105 (2013).Google Scholar
  11. 11.
    D. V. Karpov, “Minimal k-connected graphs with minimal number of vertices of degree k,” Zap. Nauchn. Semin. POMI, 427, 41–65 (2014).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.St.Petersburg Department of the Steklov Mathematical InstituteSt.Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations