Abstract
Simple drawings are drawings of graphs in which any two edges intersect at most once (either at a common endpoint or a proper crossing), and no edge intersects itself. We analyze several characteristics of simple drawings of complete multipartite graphs: which pairs of edges cross, in which order they cross, and the cyclic order around vertices and crossings, respectively. We consider all possible combinations of how two drawings can share some characteristics and determine which other characteristics they imply and which they do not imply. Our main results are that for simple drawings of complete multipartite graphs, the orders in which edges cross determine all other considered characteristics. Further, if all partition classes have at least three vertices, then the pairs of edges that cross determine the rotation system and the rotation around the crossings determine the extended rotation system. We also show that most other implications – including the ones that hold for complete graphs – do not hold for complete multipartite graphs. Using this analysis, we establish which types of isomorphisms are meaningful for simple drawings of complete multipartite graphs.
O. Aichholzer and B. Vogtenhuber partially supported by Austrian Science Fund (FWF) within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35. O. Aichholzer, B. Vogtenhuber and A. Weinberger partially supported by FWF grant W1230.
We thank the reviewers of EuroCG’21 and GD’23 for their very helpful comments.
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Aichholzer, O., Chiu, M.K., Hoang, H.P., Hoffmann, M., Kynčl, J., Maus, Y., Vogtenhuber, B., Weinberger, A.: Drawings of complete multipartite graphs up to triangle flips. In: 39th International Symposium on Computational Geometry, LIPIcs. Leibniz International Proceedings in Informatics, vol. 258, pp. 6:1–6:16. Schloss Dagstuhl. Leibniz-Zent. Inform. Wadern (2023). https://doi.org/10.4230/lipics.socg.2023.6
Aichholzer, O., García, A., Parada, I., Vogtenhuber, B., Weinberger, A.: Shooting stars in simple drawings of \(K_{m, n}\). In: Angelini, P., von Hanxleden, R. (eds.) GD 2022. LNCS, vol. 13764, pp. 49–57. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-22203-0_5
Aichholzer, O., Vogtenhuber, B., Weinberger, A.: Different types of isomorphisms of drawings of complete multipartite graphs (2023). http://arxiv.org/abs/2308.10735v1
Arroyo, A., McQuillan, D., Richter, R.B., Salazar, G.: Drawings of \(K_n\) with the same rotation scheme are the same up to triangle-flips (Gioan’s theorem). Australas. J. Combin. 67, 131–144 (2017). https://ajc.maths.uq.edu.au/pdf/67/ajc_v67_p131.pdf
Asano, K.: The crossing number of \(K_{1,3, n}\) and \(K_{2,3, n}\). J. Graph Theory 10(1), 1–8 (1986). https://doi.org/10.1002/jgt.3190100102
Brötzner, A.: Isomorphism Classes of Drawings of \(K_{3,3}\). Bachelor’s thesis, Graz University of Technology (2021)
Cardinal, J., Felsner, S.: Topological drawings of complete bipartite graphs. J. Comput. Geom. 9(1), 213–246 (2018). https://doi.org/10.20382/jocg.v9i1a7
Fabila-Monroy, R., Paul, R., Viafara-Chanchi, J., Weinberger, A.: On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs. In: Abstracts of XX Encuentros de Geometría Computacional (EGC 2023), pp. 33–36 (2023). https://egc23.web.uah.es/wp-content/uploads/2023/07/EGC2023_Booklet.pdf#page=45
Gethner, E., Hogben, L., Lidickỳ, B., Pfender, F., Ruiz, A., Young, M.: On crossing numbers of complete tripartite and balanced complete multipartite graphs. J. Graph Theory 4(84), 552–565 (2017). https://onlinelibrary.wiley.com/doi/full/10.1002/jgt.22041
Gioan, E.: Complete graph drawings up to triangle mutations. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 139–150. Springer, Heidelberg (2005). https://doi.org/10.1007/11604686_13
Gioan, E.: Complete graph drawings up to triangle mutations. Discrete Comput. Geom. 67, 985–1022 (2022). https://doi.org/10.1007/s00454-021-00339-8
Harborth, H.: Über die Kreuzungszahl vollständiger, \(n\)-geteilter Graphen. Math. Nachr. 48, 179–188 (1971). https://doi.org/10.1002/mana.19710480113
Harborth, H.: Parity of numbers of crossings for complete \(n\)-partite graphs. Mathematica Slovaca 26(2), 77–95 (1976). http://eudml.org/doc/33976
Ho, P.T.: The crossing number of \(K_{2,4, n}\). Ars Combin. 109, 527–537 (2013)
Kynčl, J.: Simple realizability of complete abstract topological graphs in P. Discrete Comput. Geom. 45(3), 383–399 (2011). https://doi.org/10.1007/s00454-010-9320-x
Kynčl, J.: Improved enumeration of simple topological graphs. Discrete Comput. Geom. 50(3), 727–770 (2013). https://doi.org/10.1007/s00454-013-9535-8
Mengersen, I.: Kreuzungsfreie Kanten in vollständigen n-geteilten Graphen. Ph.D. thesis, Technische Universität Braunschweig (1975)
Ouyang, Z., Wang, J., Huang, Y.: Two recursive inequalities for crossing numbers of graphs. Front. Math. China 12(3), 703–709 (2017). https://doi.org/10.1007/s11464-016-0618-8
Prinoth, K.: Computing exhaustive lists of complete bipartite simple drawings. Bachelor’s thesis, Graz University of Technology (2021)
Schaefer, M.: Crossing numbers of graphs. Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL (2018). https://doi.org/10.1201/9781315152394
Schaefer, M.: Taking a detour; or, Gioan’s theorem, and pseudolinear drawings of complete graphs. Discrete Comput. Geom. 66, 12–31 (2021). https://doi.org/10.1007/s00454-021-00296-2
Zarankiewicz, K.: On a problem of P. Turan concerning graphs. Fund. Math. 41, 137–145 (1954). https://doi.org/10.4064/fm-41-1-137-145
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Aichholzer, O., Vogtenhuber, B., Weinberger, A. (2023). Different Types of Isomorphisms of Drawings of Complete Multipartite Graphs. In: Bekos, M.A., Chimani, M. (eds) Graph Drawing and Network Visualization. GD 2023. Lecture Notes in Computer Science, vol 14466. Springer, Cham. https://doi.org/10.1007/978-3-031-49275-4_3
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