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Oriented Matroids

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Encyclopedia of Optimization

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Article Outline

Keywords

Historical Review

Axiom Systems for Oriented Matroids.

  Circuits and Circuit Axioms.

  Minors

  Duality.

  Chirotopes and Basis Orientations

  Vectors and Covectors

General Topics

  Directed Graphs

  Real Vectors Spaces

  Vector Configurations

  Point Configurations

  Hyperplane Arrangements.

  Topological Representation Theorem

Conclusions

See also

References

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References

  1. Bachem A, Kern W (1992) Linear programming duality: An introduction to oriented matroids. Springer, Berlin

    MATH  Google Scholar 

  2. Björner A, Las Vergnas M, Sturmfels B, White N, Ziegler GM (1993) Oriented matroids. Encycl Math Appl, vol 46. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  3. Bland RG (1977) A combinatorial abstraction of linear programming. J Combin Th B 23:33–57

    MathSciNet  MATH  Google Scholar 

  4. Bland RG (1977) New finite pivoting rules for the simplex method. Math Oper Res 2:103–107

    MathSciNet  MATH  Google Scholar 

  5. Bland RG, Las Vergnas M (1978) Orientability of matroids. J Combin Th B 23:94–123

    MathSciNet  Google Scholar 

  6. Bland RG, Las Vergnas M (1979) Minty colorings and orientations of matroids. Ann New York Acad Sci 319:86–92

    MathSciNet  Google Scholar 

  7. Edmonds J (1965) Lehman's switching game and a theorem of Tutte and Nash–Williams. J Res Nat Bureau Standards (B) 69:73–77

    MathSciNet  MATH  Google Scholar 

  8. Edmonds J (1965) Maximum matching and a polyhedron with {0, 1} vertices. J Res Nat Bureau Standards (B) 69:125–130

    MathSciNet  MATH  Google Scholar 

  9. Edmonds J (1965) Minimum partition of a matroid into independent subsets. J Res Nat Bureau Standards (B) 69:67–72

    MathSciNet  MATH  Google Scholar 

  10. Edmonds J (1965) Paths, trees, and flowers. Canad J Math 17:449–467

    MathSciNet  MATH  Google Scholar 

  11. Edmonds J (1967) Optimum branchings. J Res Nat Bureau Standards (B) 71:233–240

    MathSciNet  MATH  Google Scholar 

  12. Edmonds J (1967) Systems of distinct representatives and linear algebra. J Res Nat Bureau Standards (B) 71:241–245

    MathSciNet  MATH  Google Scholar 

  13. Edmonds J (1970) Submodular functions, matroids, and certain polyhedra. In: Guy R, Hanani H, Sauer N, Schönheim J (eds) Combinatorial Structures and Their Applications. Gordon and Breach, New York

    Google Scholar 

  14. Edmonds J, Mandel A (1982) Topology of oriented matroids. PhD Thesis of A. Mandel, University Waterloo

    Google Scholar 

  15. Folkman J, Lawrence J (1978) Oriented matroids. J Combin Th B 25:199–236

    MathSciNet  MATH  Google Scholar 

  16. Fulkerson DR (1968) Networks, frames, blocking systems. In: Dantzig GB, Veinott AF (eds) Mathematics of the Decision Systems, Part I. Lect Appl Math., vol 2. Amer. Math. Soc., Providence, pp 303–334

    Google Scholar 

  17. Lawrence J (1975) Oriented matroids. PhD Thesis, University Washington, Seattle

    Google Scholar 

  18. Lawrence J (1982) Oriented matroids and multiply ordered sets. Linear Alg & Its Appl 48:1–12

    MathSciNet  MATH  Google Scholar 

  19. Minty GJ (1966) On the axiomatic foundations of the theories of directed linear graphs, electrical networks, and network-programming. J Math Mechanics 15:485–520

    MathSciNet  MATH  Google Scholar 

  20. Oxley JG (1992) Matroid theory. Oxford University Press, Oxford

    MATH  Google Scholar 

  21. Rockafellar RT (1969) The elementary vectors of a subspace of Rn. In: Combinatorial Mathematics and Its Applications (Proc. Chapel Hill Conf.). University North Carolina Press, Chapel Hill, pp 104–127

    Google Scholar 

  22. Tutte WT (1958) A homotopy theorem for matroids I–II. Trans Amer Math Soc 88:144–160; 161–174

    MathSciNet  Google Scholar 

  23. Tutte WT (1959) Matroids and graphs. Trans Amer Math Soc 90:527–552

    MathSciNet  MATH  Google Scholar 

  24. Tutte WT (1960) An algorithm for determining wheter a given binary matroid is graphic. Proc Amer Math Soc 11:905–917

    MathSciNet  Google Scholar 

  25. Tutte WT (1961) On the problem of decomposing a graph into n connected factors. J London Math Soc 36:221–230

    MathSciNet  MATH  Google Scholar 

  26. Tutte WT (1964) From matrices to graphs. Canad J Math 16:108–127

    MathSciNet  MATH  Google Scholar 

  27. Tutte WT (1965) Lectures on matroids. J Res Nat Bureau Standards (B) 69:1–47

    MathSciNet  MATH  Google Scholar 

  28. Tutte WT (1966) Connectivity in graphs. University Toronto Press, Toronto

    MATH  Google Scholar 

  29. Tutte WT (1966) Connectivity in matroids. Canad J Math 18:1301–1324

    MathSciNet  MATH  Google Scholar 

  30. Tutte WT (1971) Introduction to the theory of matroids. Amer. Elsevier, New York

    MATH  Google Scholar 

  31. Las Vergnas M (1975) Coordinatizable strong maps of matroids (preprint)

    Google Scholar 

  32. Las Vergnas M (1975) Matroides orientables. CR Acad Sci Paris Ser A 280:61–64

    MathSciNet  MATH  Google Scholar 

  33. Las Vergnas M (1977) Acyclic and totally cyclic orientations of combinatorial geometries. Discret Math 20:51–61

    MathSciNet  Google Scholar 

  34. Las Vergnas M (1978) Bases in oriented matroids. J Combin Th B 25:283–289

    MathSciNet  MATH  Google Scholar 

  35. Whitney H (1935) On the abstract properties of linear dependence. Amer J Math 57:509–533

    MathSciNet  Google Scholar 

  36. Ziegler GM (1998) Oriented matroids today. Electronic J Combin 3

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dip. Mat. e Inform., University Salerno, Baronissi, Italy

    Paola Festa

Authors
  1. Paola Festa
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Editor information

Editors and Affiliations

  1. Department of Chemical Engineering, Princeton University, Princeton, NJ, 08544-5263, USA

    Christodoulos A. Floudas

  2. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Panos M. Pardalos

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© 2008 Springer-Verlag

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Festa, P. (2008). Oriented Matroids . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_493

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