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Betweenness and Comparability Obtained from Binary Relations

  • Ivo Düntsch
  • Alasdair Urquhart
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4136)

Abstract

We give a brief overview of the axiomatic development of betweenness relations, and investigate the connections between these and comparability graphs. Furthermore, we characterize betweenness relations induced by reflexive and antisymmetric binary relations, thus generalizing earlier results on partial orders. We conclude with a sketch of the algorithmic aspects of recognizing induced betweenness relations.

Keywords

Partial Order Binary Relation Axiom System Interval Graph Comparability Graph 
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References

  1. 1.
    Altwegg, M.: Zur Axiomatik der teilweise geordneten Mengen. Commentarii Mathematici Helvetici 24, 149–155 (1950)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Birkhoff, G.: Lattice Theory, 2nd revised edn. American Mathematical Society (1948)Google Scholar
  3. 3.
    Düvelmeyer, N., Wenzel, W.: A characterization of ordered sets and lattices via betweenness relations. Resultate der Mathematik 46, 237–250 (2004)MATHGoogle Scholar
  4. 4.
    Ghouila-Houri, A.: Caractérisation des graphes non orientés dont on peut orienter les arêtes de manière à obtenir le graphe d’une relation d’ordre. C.R. Acad. Sci. Paris, pp. 1370–1371 (1962)Google Scholar
  5. 5.
    Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and of interval graphs. Canadian Journal of Mathematics 16, 539–548 (1964)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Golumbic, M.C.: Comparability graphs and a new matroid. Journal of Combinato-rial Theory 22, 68–90 (1977)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Golumbic, M.C.: The complexity of comparability graph recognition and coloring. Computing 18, 199–208 (1977)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Golumbic, M.C.: Algorithmic graph theory and perfect graphs. Academic Press, New York (1980)MATHGoogle Scholar
  9. 9.
    Huntington, E.V.: A new set of postulates for betweenness, with proof of complete independence. Trans. Am Math. Soc. 26, 257–282 (1924)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Huntington, E.V., Kline, J.R.: Set of independent postulates for betweenness. Trans. Am. Math. Soc. 18, 301–325 (1917)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Kelly, D.: Comparability graphs. In: Rival, I. (ed.) Graphs and Order, pp. 3–40. D. Reidel Publishing Company (1985)Google Scholar
  12. 12.
    Menger, K.: Untersuchungen über die allgemeine Metrik. Mathematische Annalen 100, 75–163 (1928)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Möhring, R.H.: Algorithmic aspects of comparability graphs and interval graphs. In: Rival, I. (ed.) Graphs and Order, pp. 41–101. D. Reidel Publishing Company (1985)Google Scholar
  14. 14.
    Pitcher, E., Smiley, M.F.: Transitivities of betweenness. Trans. Am. Math. Soc. 52, 95–114 (1942)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Sholander, M.: Trees, lattices, order and betweenness. Proc. Am. Math. Soc. 3(3), 369–381 (1952)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Tarski, A., Givant, S.: Tarski’s system of geometry. The Bulletin of Symbolic Logic 5(2), 175–214 (1998)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ivo Düntsch
    • 1
  • Alasdair Urquhart
    • 2
  1. 1.Department of Computer ScienceBrock UniversitySt. CatharinesCanada
  2. 2.Department of PhilosophyUniversity of TorontoTorontoCanada

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