On semi-simple drawings of the complete graph

Oswin Aichholzer, Florian Ebenführer, Irene Parada, Alexander Pilz, Birgit Vogtenhuber

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In this work we study rotation systems and semi-simple drawings of $K_n$. A simple drawing of a graph is a drawing in which every pair of edges intersects in at most one point. In a semi-simple drawing, edge pairs might intersect in multiple points, but incident edges only intersect in their common endpoint. A rotation system is called (semi-)realizable if it can be realized with a (semi-)simple drawing. It is known that a rotation system is realizable if and only if all its 5-tuples are realizable. For the problem of characterizing semi-realizability, we present a rotation system with six vertices that is not semi-realizable, although all its 5-tuples are semi-realizable. Moreover, by an exhaustive computer search, we show that also for seven vertices there exist minimal not semi-realizable rotation systems (that is, rotation systems in which all proper sub-rotation systems are semi-realizable). This indicates that checking semi-realizability is harder than checking realizability. Finally we show that for semi-simple drawings, generalizations of Conway's Thrackle Conjecture and the conjecture on the existence of plane Hamiltonian cycles do not hold.
Original languageEnglish
Title of host publicationProc. XVII Encuentros de Geometría Computacional
Place of PublicationAlicante, Spain
Number of pages4
Publication statusPublished - 2017
EventXVII Spanish Meeting on Computational Geometry - , Spain
Duration: 26 Jun 201728 Jun 2017


ConferenceXVII Spanish Meeting on Computational Geometry
Abbreviated titleEGC 2017
Internet address

Cite this

Aichholzer, O., Ebenführer, F., Parada, I., Pilz, A., & Vogtenhuber, B. (2017). On semi-simple drawings of the complete graph. In Proc. XVII Encuentros de Geometría Computacional (pp. 25-28). Alicante, Spain.