On Tope Graphs of Complexes of Oriented Matroids

  • 21 Accesses


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.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


  1. 1.

    Albenque, M., Knauer, K.: Convexity in partial cubes: the hull number. Discrete Math. 339(2), 866–876 (2016)

  2. 2.

    Bandelt, H.-J.: Graphs with intrinsic \(S_3\) convexities. J. Graph Theory 13(2), 215–227 (1989)

  3. 3.

    Bandelt, H.-J., Chepoi, V., Knauer, K.: COMs: complexes of oriented matroids. J. Comb. Theory Ser. A 156, 195–237 (2018)

  4. 4.

    Baum, A., Zhu, Y.: The axiomatization of affine oriented matroids reassessed. J. Geom. 109(1), 11 (2018)

  5. 5.

    Berman, A., Kotzig, A.: Cross-cloning and antipodal graphs. Discrete Math. 69(2), 107–114 (1988)

  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)

  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)

  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)

  9. 9.

    Chepoi, V.: \(d\)-Convex sets in graphs. Dissertation, Moldova State University (1986)

  10. 10.

    Chepoi, V.: Separation of two convex sets in convexity structures. J. Geom. 50(1–2), 30–51 (1994)

  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)

  13. 13.

    Cordovil, R.: A combinatorial perspective on the non-Radon partitions. J. Comb. Theory Ser. A 38(1), 38–47 (1985)

  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)

  15. 15.

    Delucchi, E., Knauer, K.: Finitary affine oriented matroids (in preparation)

  16. 16.

    Desgranges, R., Knauer, K.: A correction of a characterization of planar partial cubes. Discrete Math. 340(6), 1151–1153 (2017)

  17. 17.

    Djoković, D.Ž.: Distance-preserving subgraphs of hypercubes. J. Comb. Theory Ser. B 14, 263–267 (1973)

  18. 18.

    Dress, A.W.M., Scharlau, R.: Gated sets in metric spaces. Aequationes Math. 34(1), 112–120 (1987)

  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)

  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)

  21. 21.

    Eppstein, D., Falmagne, J.-C., Ovchinnikov, S.: Media Theory. Springer, Berlin (2008)

  22. 22.

    Fukuda, K.: Lecture Notes on Oriented Matroids and Geometric Computation. ETH Zürich, Zürich (2004)

  23. 23.

    Fukuda, K., Handa, K.: Antipodal graphs and oriented matroids. Discrete Math. 111(1–3), 245–256 (1993)

  24. 24.

    Fukuda, K., Saito, S., Tamura, A.: Combinatorial face enumeration in arrangements and oriented matroids. Discrete Appl. Math. 31(2), 141–149 (1991)

  25. 25.

    Graham, R.L., Pollak, H.O.: On the addressing problem for loop switching. Bell System Tech. J. 50, 2495–2519 (1971)

  26. 26.

    Handa, K.: A characterization of oriented matroids in terms of topes. Eur. J. Comb. 11(1), 41–45 (1990)

  27. 27.

    Handa, K.: Topes of oriented matroids and related structures. Publ. Res. Inst. Math. Sci. 29(2), 235–266 (1993)

  28. 28.

    Karlander, J.: A characterization of affine sign vector Systems. PhD Thesis, Kungliga Tekniska Högskolan Stockholm (1992)

  29. 29.

    Klavžar, S., Shpectorov, S.: Convex excess in partial cubes. J. Graph Theory 69(4), 356–369 (2012)

  30. 30.

    Lawrence, J.: Lopsided sets and orthant-intersection by convex sets. Pac. J. Math. 104(1), 155–173 (1983)

  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)

  32. 32.

    Ovchinnikov, S.: Graphs and Cubes. Universitext. Springer, Berlin (2011)

  33. 33.

    Peterin, I.: A characterization of planar partial cubes. Discrete Math. 308(24), 6596–6600 (2008)

  34. 34.

    Polat, N.: On bipartite graphs whose interval space is a closed join space. J. Geom. 108(2), 719–741 (2017)

  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)

  36. 36.

    Winkler, P.M.: Isometric embedding in products of complete graphs. Discrete Appl. Math. 7(2), 221–225 (1984)

Download references


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.

Author information

Correspondence to Kolja Knauer.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Editor in Charge: Kenneth Clarkson

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Knauer, K., Marc, T. On Tope Graphs of Complexes of Oriented Matroids. Discrete Comput Geom (2019).

Download citation


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

Mathematics Subject Classification

  • 05C75
  • 05C12