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.
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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.
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Knauer, K., Marc, T. On Tope Graphs of Complexes of Oriented Matroids. Discrete Comput Geom (2019). https://doi.org/10.1007/s00454-019-00111-z
- Tope graphs
- Complexes of oriented matroids
- Oriented matroids
- Lopsided sets
- Partial cubes
Mathematics Subject Classification