On Tope Graphs of Complexes of Oriented Matroids

  • Kolja KnauerEmail author
  • Tilen Marc


We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third characterization in terms of zone graphs of tope graphs. Further corollaries include a characterization of topes of oriented matroids due to da Silva, another one of Handa, a characterization of lopsided systems due to Lawrence, and an intrinsic characterization of tope graphs of affine oriented matroids. Moreover, we obtain purely graph theoretic polynomial time recognition algorithms for tope graphs of the above and a finite list of excluded partial cube minors for the bounded rank case. In particular, our results answer a relatively long-standing open question in oriented matroids and can be seen as identifying the theory of (complexes of) oriented matroids as a part of metric graph theory. Another consequence is that all finite Pasch graphs are tope graphs of complexes of oriented matroids, which confirms a conjecture of Chepoi and the two authors.


Tope graphs Complexes of oriented matroids Oriented matroids Lopsided sets Partial cubes 

Mathematics Subject Classification

05C75 05C12 



We wish to thank Emanuele Delucchi and Yida Zhu for discussions on AOMs, Ilda da Silva for insights on OMs, Hans-Jürgen Bandelt and Victor Chepoi for several fruitful discussions on COMs and their tope graphs, and Matjaž Kovše for being part of the very first sessions on tope graphs. The first author was supported by Grants ANR-16-CE40-0009-01, ANR-17-CE40-0015, and ANR-17-CE40-0018, the second author by Grants P1-0297 and J1-9109. Finally, we wish to thank the referees for comments that clearly improved the quality of this paper.


  1. 1.
    Albenque, M., Knauer, K.: Convexity in partial cubes: the hull number. Discrete Math. 339(2), 866–876 (2016)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bandelt, H.-J.: Graphs with intrinsic \(S_3\) convexities. J. Graph Theory 13(2), 215–227 (1989)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bandelt, H.-J., Chepoi, V., Knauer, K.: COMs: complexes of oriented matroids. J. Comb. Theory Ser. A 156, 195–237 (2018)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Baum, A., Zhu, Y.: The axiomatization of affine oriented matroids reassessed. J. Geom. 109(1), 11 (2018)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Berman, A., Kotzig, A.: Cross-cloning and antipodal graphs. Discrete Math. 69(2), 107–114 (1988)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Björner, A.: Topological methods. In: Graham, R.L., et al. (eds.) Handbook of Combinatorics, vol. 1, 2, pp. 1819–1872. Elsevier, Amsterdam (1995)Google Scholar
  7. 7.
    Björner, A., Edelman, P.H., Ziegler, G.M.: Hyperplane arrangements with a lattice of regions. Discrete Comput. Geom. 5(3), 263–288 (1990)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids. Encyclopedia of Mathematics and its Applications, vol. 46, 2nd edn. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  9. 9.
    Chepoi, V.: \(d\)-Convex sets in graphs. Dissertation, Moldova State University (1986)Google Scholar
  10. 10.
    Chepoi, V.: Separation of two convex sets in convexity structures. J. Geom. 50(1–2), 30–51 (1994)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chepoi, V., Knauer, K., Marc, T.: Partial cubes without \(Q_3^-\) minors (2016). arXiv:1606.02154
  12. 12.
    Chepoj, V.: Isometric subgraphs of Hamming graphs and \(d\)-convexity. Cybernetics 24, 6–11 (1988)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cordovil, R.: A combinatorial perspective on the non-Radon partitions. J. Comb. Theory Ser. A 38(1), 38–47 (1985)MathSciNetzbMATHGoogle Scholar
  14. 14.
    da Silva, I.P.F.: Axioms for maximal vectors of an oriented matroid: a combinatorial characterization of the regions determined by an arrangement of pseudohyperplanes. Eur. J. Comb. 16(2), 125–145 (1995)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Delucchi, E., Knauer, K.: Finitary affine oriented matroids (in preparation)Google Scholar
  16. 16.
    Desgranges, R., Knauer, K.: A correction of a characterization of planar partial cubes. Discrete Math. 340(6), 1151–1153 (2017)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Djoković, D.Ž.: Distance-preserving subgraphs of hypercubes. J. Comb. Theory Ser. B 14, 263–267 (1973)MathSciNetGoogle Scholar
  18. 18.
    Dress, A.W.M., Scharlau, R.: Gated sets in metric spaces. Aequationes Math. 34(1), 112–120 (1987)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Eppstein, D.: Recognizing partial cubes in quadratic time. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pp. 1258–1266. ACM, New York (2008)Google Scholar
  20. 20.
    Eppstein, D.: Isometric diamond subgraphs. In: Tollis, I.G., Patrignani, M. (eds.) Graph Drawing. Lecture Notes in Computer Science, vol. 5417, pp. 384–389. Springer, Berlin (2009)Google Scholar
  21. 21.
    Eppstein, D., Falmagne, J.-C., Ovchinnikov, S.: Media Theory. Springer, Berlin (2008)zbMATHGoogle Scholar
  22. 22.
    Fukuda, K.: Lecture Notes on Oriented Matroids and Geometric Computation. ETH Zürich, Zürich (2004)Google Scholar
  23. 23.
    Fukuda, K., Handa, K.: Antipodal graphs and oriented matroids. Discrete Math. 111(1–3), 245–256 (1993)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Fukuda, K., Saito, S., Tamura, A.: Combinatorial face enumeration in arrangements and oriented matroids. Discrete Appl. Math. 31(2), 141–149 (1991)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Graham, R.L., Pollak, H.O.: On the addressing problem for loop switching. Bell System Tech. J. 50, 2495–2519 (1971)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Handa, K.: A characterization of oriented matroids in terms of topes. Eur. J. Comb. 11(1), 41–45 (1990)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Handa, K.: Topes of oriented matroids and related structures. Publ. Res. Inst. Math. Sci. 29(2), 235–266 (1993)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Karlander, J.: A characterization of affine sign vector Systems. PhD Thesis, Kungliga Tekniska Högskolan Stockholm (1992)Google Scholar
  29. 29.
    Klavžar, S., Shpectorov, S.: Convex excess in partial cubes. J. Graph Theory 69(4), 356–369 (2012)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Lawrence, J.: Lopsided sets and orthant-intersection by convex sets. Pac. J. Math. 104(1), 155–173 (1983)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Marc, T.: There are no finite partial cubes of girth more than 6 and minimum degree at least 3. Eur. J. Comb. 55, 62–72 (2016)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Ovchinnikov, S.: Graphs and Cubes. Universitext. Springer, Berlin (2011)zbMATHGoogle Scholar
  33. 33.
    Peterin, I.: A characterization of planar partial cubes. Discrete Math. 308(24), 6596–6600 (2008)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Polat, N.: On bipartite graphs whose interval space is a closed join space. J. Geom. 108(2), 719–741 (2017)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Vapnik, V.N., Chervonenkis, A. Ya.: On the uniform convergence of relative frequencies of events to their probabilities. In: Vovk, V., et al. (eds.) Measures of Complexity, pp. 11–30. Springer, Cham (2015)Google Scholar
  36. 36.
    Winkler, P.M.: Isometric embedding in products of complete graphs. Discrete Appl. Math. 7(2), 221–225 (1984)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Aix-Marseille UniversiteMarseilleFrance

Personalised recommendations