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.
Similar content being viewed by others
References
J. Edmonds, Edge-disjoint branchings, in:Combinatorial Algorithms, (R. Rustin, ed.) Algorithmics Press, New York, 1973, 91–96.
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.
R. P. Gupta, A decomposition theorem for bipartite graphs, in:Theory of Graphs (P. Rosenstiehl, ed.) Gordon and Breach, New York, 1967, 135–138.
L. Lovász, On two minimax theorems in graph theory,J. Combinatorial Theory (B) 21 (1976), 96–103.
C. L. Lucchesi andD. H. Younger, A minimax relation for directed graphs,J. London Math. Soc. (2) 17 (1978), 369–374.
A. Schrijver, A counterexample to a conjecture of Edmonds and Giles,Discrete Math. 32 (1980), 213–214.
A. Schrijver, Min-max relations for directed graphs,Annals of Discrete Math. 16 (1982), 261–280.
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.
D. H. Younger, From shortest paths to directed cut transversals,Annals of the New York Academy of Sciences,319 (1979), 555–564.
D. H. Younger, Maximum families of disjoint directed cut sets,in: Recent Progress in Combinatorics (W. T. Tutte, ed.) Academic Press, 1969, 329–333.
P. Feofiloff,Disjoint Transversals of Directed Coboundaries, Ph. D. Thesis, University of Waterloo, 1983.
P. D. Seymour, The matroids with the max-flow min-cut property,J. Combinatorial Theory (B) 23 (1977), 189–222.
Author information
Authors and Affiliations
Rights 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
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02579302