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Menger and könig systems

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Theory and Applications of Graphs

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 642))

Abstract

A system of sets is called a Menger system or a König system if it has properties that were originally proved to hold for certain special systems by Menger and by König, respectively, in the theorems that customarily bear their names. In this survey paper, these properties are specified, and the relationship between them is described. Some known theorems about them are stated. Some systems of sets of edges of graphs with these properties are exhibited. The main (and very substantial) omission from this survey is any mention of rational analogues of these properties and the connection with polyhedra and linear programming, for which see, for example, [2–5, 10, 16, 17] and references cited therein.

This paper was written while the author was visiting the University of Calgary.

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References

  1. C. Berge, Graphs and Hypergraphs. North-Holland, Amsterdam (1973).

    MATH  Google Scholar 

  2. J. Edmonds and D. R. Fulkerson, Bottleneck extrema. J. Combinatorial Theory 8 (1970), 299–306.

    Article  MathSciNet  Google Scholar 

  3. D. R. Fulkerson, Networks, frames, blocking systems. Mathematics of the Decision Sciences Part I (G. B. Dantzig and A. F. Veinott, eds). American Mathematics Society Lectures in Applied Mathematics 11 (1968), 303–334.

    MathSciNet  Google Scholar 

  4. D. R. Fulkerson, Blocking polyhedra. Graph Theory and its Applications (B. Harris, ed.). Academic Press, New York (1970), 93–112.

    Google Scholar 

  5. D. R. Fulkerson, Antiblocking polyhedra. J. Combinatorial Theory Ser. B 12 (1972), 50–71.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  7. L.Las Vergnas, Matroides orientables. Duplicated typescript. 1974.

    Google Scholar 

  8. L. Lovasz, Normal hypergraphs and the perfect graph conjecture. Discrete Math. 2 (1972), 253–267.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Lovasz, Minimax theorems for hypergraphs. Hypergraph Seminar (A. Dold and B. Eckmann, eds.) Springer Lecture Notes in Mathematics 211 (1974), 111–126.

    Article  MathSciNet  Google Scholar 

  10. B. Rathschild and A. Whinston, On two commodity network flows. Operations Reserach 14 (1966), 377–387.

    Article  Google Scholar 

  11. L. Lovasz, Two minimax theorems in graph theory. J. Combinatorial Theory Ser. B, to appear.

    Google Scholar 

  12. C. L. Lucchesi and D. H. Younger, A minimax theorem for directed graphs. To appear.

    Google Scholar 

  13. T. Gallai, Über reguläre Kettengruppen. Acta Math. Acad. Sci. Hungar 10 (1959), 227–240.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. D. Seymour, The matroids with the max-flow min-cut property. J. Combinatorial Theory Ser. B, to appear.

    Google Scholar 

  15. D. R. Woodall, A note on rank functions and integer programming. Discrete Math., submitted (1974).

    Google Scholar 

  16. D. R. Woodall, Applications of polymatroids and linear programming to transversals and graphs. Combinatorics (T. P. McDonough and V. C. Mavron, eds). London Mathematicsl Society Lecture Note Series 13, Cambridge University Press (1974), 195–200.

    Google Scholar 

  17. D. R. Woodall, Applications of polymatroids and linear programming to transversals and graphs. Unpublished preprint (contains the proofs of results in [15]).

    Google Scholar 

  18. K. Vidyasankar and D. H. Younger, A minimax equality related to the longest directed path in an acyclic graph. Canad. J. Math. 27 (1975), 348–351.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Edmonds, Edgepdisjoint branchings. Combinatorial Plgonthoms (R. Rustin, ed.). Algorithmic Press, New York (1973), 91–96.

    Google Scholar 

  20. D. R. Fulkerson, Packing weighted cuts in rooted directed graphs, Math. Programming 6 (1974), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Nottingham, Nottingham, England, NG7 2RD

    D. R. Woodall

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  1. D. R. Woodall
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Western Michigan University, Kalamazoo, MI, 49008, USA

    Yousef Alavi  & Don R. Lick  & 

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© 1978 Springer-Verlag Berlin Heidelberg

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Woodall, D.R. (1978). Menger and könig systems. In: Alavi, Y., Lick, D.R. (eds) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol 642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0070416

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  • DOI: https://doi.org/10.1007/BFb0070416

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08666-6

  • Online ISBN: 978-3-540-35912-8

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