Directed cut transversal packing for source-sink connected graphs

Combinatorica volume 7, pages255–263(1987)Cite this article

Abstract

Concerning the conjecture that in every directed graph, a maximum packing of directed cut transversals is equal in cardinality to a minimum directed cut, a proof is given for the side coboundaries of a graph. This case includes, and is essentially equivalent to, all source-sink connected graphs, for which Schrijver has given a proof. The method used here first reduces the assertion to a packing theorem for bi-transversals. A packing of bi-transversals of the required size is constructed one edge at a time, by maintaining a Hall-like feasibility condition as each edge is added.

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

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

References

  1. [1]

    J. Edmonds, Edge-disjoint branchings, in:Combinatorial Algorithms, (R. Rustin, ed.) Algorithmics Press, New York, 1973, 91–96.

    Google Scholar 

  2. [2]

    J. Edmonds andR. Giles, A min-max relation for submodular functions on graphs,in: Studies in integer programming (P. L. Hammer, et al, eds.) Annals of Discrete Math.1 (1977), 185–204.

  3. [3]

    R. P. Gupta, A decomposition theorem for bipartite graphs, in:Theory of Graphs (P. Rosenstiehl, ed.) Gordon and Breach, New York, 1967, 135–138.

    Google Scholar 

  4. [4]

    L. Lovász, On two minimax theorems in graph theory,J. Combinatorial Theory (B) 21 (1976), 96–103.

    MATH  Article  Google Scholar 

  5. [5]

    C. L. Lucchesi andD. H. Younger, A minimax relation for directed graphs,J. London Math. Soc. (2) 17 (1978), 369–374.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    A. Schrijver, A counterexample to a conjecture of Edmonds and Giles,Discrete Math. 32 (1980), 213–214.

    MATH  MathSciNet  Google Scholar 

  7. [7]

    A. Schrijver, Min-max relations for directed graphs,Annals of Discrete Math. 16 (1982), 261–280.

    MATH  MathSciNet  Google Scholar 

  8. [8]

    D. R. Woodall, Menger and König systems,in: Theory and Applications of Graphs (Proceedings of the Western Michigan Graph Theory Conference 1976) Springer-Verlag Lecture Notes in Math.642 (1978), 620–635.

  9. [9]

    D. H. Younger, From shortest paths to directed cut transversals,Annals of the New York Academy of Sciences,319 (1979), 555–564.

    Article  MathSciNet  Google Scholar 

  10. [10]

    D. H. Younger, Maximum families of disjoint directed cut sets,in: Recent Progress in Combinatorics (W. T. Tutte, ed.) Academic Press, 1969, 329–333.

  11. [11]

    P. Feofiloff,Disjoint Transversals of Directed Coboundaries, Ph. D. Thesis, University of Waterloo, 1983.

  12. [12]

    P. D. Seymour, The matroids with the max-flow min-cut property,J. Combinatorial Theory (B) 23 (1977), 189–222.

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

  1. Department of Combinatorics and Optimatization, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    P. Feofiloff & D. H. Younger

Authors
  1. P. Feofiloff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. H. Younger
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Feofiloff, P., Younger, D.H. Directed cut transversal packing for source-sink connected graphs. Combinatorica 7, 255–263 (1987). https://doi.org/10.1007/BF02579302

Download citation

AMS subject classification (1980)

  • 05 C 20