On digraphs with no two disjoint directed cycles

Abstract

We obtain a result on configurations in 2-connected digraphs with no two disjoint dicycles. We derive various consequences, for example a short proof of the characterization of the minimal digraphs having no vertex meeting all dicycles and a polynomially bounded algorithm for finding a dicycle through any pair of prescribed arcs in a digraph with no two disjoint dicycles, a problem which is NP-complete for digraphs in general.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    E. Allender, On the number of cycles possible in digraphs with large girth,Discrete Appl. Math.,10 (1985), 211–225.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    J.-C. Bermond andC. Thomassen, Cycles in digraphs—a survey,J. Graph Theory,5 (1981), 1–43.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    G. A. Dirac, Some results concerning the structure of graphs,Canad. Math. Bull.,6 (1963) 183–210.

    MATH  MathSciNet  Google Scholar 

  4. [4]

    Z. Ésik, On cycles of directed and undirected graphs,to appear.

  5. [5]

    S. Fortune, J. Hopcroft andJ. Wyllie, The directed subgraph homeomorphism problem,J. Theoret. Comput. Sci.,10 (1980), 111–121.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    T. Gallai, Problem 6in: Theory of Graphs, Proc. Colloq. Tihany 1966, Academic Press, New York (1968), 362.

    Google Scholar 

  7. [7]

    S. R. Kosaraju, On independent circuits of a digraph.J. Graph Theory,1 (1977) 379–382.

    MATH  MathSciNet  Google Scholar 

  8. [8]

    A. V. Kostochka, A problem on directed graphs (in Russian with English summary),Acta Cybernet.,6 (1983), 89–91.

    MATH  MathSciNet  Google Scholar 

  9. [9]

    L. Lovász, On graphs not containing independent circuits (in Hungarian),Mat. Lapok,16 (1965), 289–299.

    MATH  MathSciNet  Google Scholar 

  10. [10]

    C. L. Lucchesi andD. H. Younger, A minimax theorem for directed graphs,J. Lond. Math. Soc.,17 (1978), 369–374.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    C. Thomassen, The 2-linkage problem for acyclic digraphs,Discrete Math.,55 (1985), 73–87.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    C. Thomassen, Paths, circuits and subdivisions,in: Selected Topics in Graph Theory III (L. W. Beineke and R. J. Wilson eds.) Academic Press,to appear.

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Thomassen, C. On digraphs with no two disjoint directed cycles. Combinatorica 7, 145–150 (1987). https://doi.org/10.1007/BF02579210

Download citation

AMS subject classification (1980)

  • 05 C 20
  • 05 C 38