, Volume 28, Issue 1, pp 89–97 | Cite as

Strict Betweennesses Induced by Posets as well as by Graphs

  • Dieter RautenbachEmail author
  • Philipp Matthias Schäfer


For a finite poset P = (V, ≤ ), let \({\cal B}_s(P)\) consist of all triples (x,y,z) ∈ V 3 such that either x < y < z or z < y < x. Similarly, for every finite, simple, and undirected graph G = (V,E), let \({\cal B}_s(G)\) consist of all triples (x,y,z) ∈ V 3 such that y is an internal vertex on an induced path in G between x and z. The ternary relations \({\cal B}_s(P)\) and \({\cal B}_s(G)\) are well-known examples of so-called strict betweennesses. We characterize the pairs (P,G) of posets P and graphs G on the same ground set V which induce the same strict betweenness relation \({\cal B}_s(P)={\cal B}_s(G)\).


Poset Graph Induced path Betweenness Convexity 

Mathematics Subject Classifications (2010)

05C99 06A06 52A01 52A37 


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  1. 1.
    Altwegg, M.: Zur Axiomatik der teilweise geordneten Mengen. Comment. Math. Helv. 24, 149–155 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Birkhoff, G.: Lattice Theory, Revised Edn., 285 pp. American Mathematical Society Colloquium Publications, 25. American Mathematical Society (AMS) VIII, New York (1948)Google Scholar
  3. 3.
    Chvátal, V.: Sylvester–Gallai theorem and metric betweenness. Discrete Comput. Geom. 31, 175–195 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chvátal, V.: Antimatroids, betweenness, convexity. In: Cook, W.J., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization (Bonn 2008). Springer, Berlin (2009)Google Scholar
  5. 5.
    Cox, D., Burmeister, M., Price, E., Kim, S., Myers, R.: Radiation hybrid mapping: a somatic cell genetic method for constructing high-resolution maps of mammalian chromosomes. Science 250(4978), 245–250 (1990)CrossRefGoogle Scholar
  6. 6.
    Duchet, P.: Convex sets in graphs. II: minimal path convexity. J. Comb. Theory, Ser. B 44, 307–316 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Düntsch, I., Urquhart, A.: Betweenness and comparability obtained from binary relations. Lect. Notes Comput. Sci. 4136, 148–161 (2006)CrossRefGoogle Scholar
  8. 8.
    Goss, S., Harris, H.: New methods for mapping genes in human chromosomes. Nature 255, 680–684 (1975)CrossRefGoogle Scholar
  9. 9.
    Gutin, G., Kim, E.J., Mnich, M., Yeo, A.: Ordinal embedding relaxations parameterized above tight lower bound. Preprint, arXiv:0907.5427v2
  10. 10.
    Haas, R., Hoffmann, M.: Chordless paths through three vertices. Theor. Comp. Sci. 351, 360–371 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Huntington, E.V., Kline, J.R.: Sets of independent postulates for betweenness. American M. S. Trans. 18, 301–325 (1916)MathSciNetGoogle Scholar
  12. 12.
    Jamison-Waldner, R.E.: A perspective on abstract convexity: classifying alignments by varieties. Lect. Notes Pure Appl. Math. 76, 113–150 (1982)MathSciNetGoogle Scholar
  13. 13.
    Krokhin, A., Jeavons, P., Jonsson, P.: Constraint satisfaction problems on intervals and lengths. SIAM J. Discrete Math. 17, 453–477 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lihová, J.: Strict order-betweennesses. Acta Univ. M. Belii, Ser. Math. 8, 27–33 (2000)zbMATHGoogle Scholar
  15. 15.
    Menger, K.: Untersuchungen über allgemeine metrik. Math. Ann. 100, 75–163 (1928)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Opatrny, J.: Total ordering problem. SIAM J. Comput. 8, 111–114 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Sholander, M.: Trees, lattices, order, and betweenness. Proc. Amer. Math. Soc. 3, 369–381 (1952)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Institut für MathematikIlmenauGermany

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