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Simplifying Non-simple Fan-Planar Drawings

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Graph Drawing and Network Visualization (GD 2021)

Abstract

A drawing of a graph is fan-planar if the edges intersecting a common edge a share a vertex A on the same side of a. More precisely, orienting e arbitrarily and the other edges towards A results in a consistent orientation of the crossings. So far, fan-planar drawings have only been considered in the context of simple drawings, where any two edges share at most one point, including endpoints. We show that every non-simple fan-planar drawing can be redrawn as a simple fan-planar drawing of the same graph while not introducing additional crossings. Combined with previous results on fan-planar drawings, this yields that n-vertex-graphs having such a drawing can have at most 6.5n edges and that the recognition of such graphs is NP-hard. We thereby answer an open problem posed by Kaufmann and Ueckerdt in 2014.

This work was initiated at the \(5^{th}\) DACH Workshop on Arrangements and Drawings, which was conducted online, via gathertown, in March 2021. The authors thank the organizers of the workshop for inviting us and providing a productive working atmosphere. B. K. was partially supported by DFG project WO 758/11-1. M. M. R. is supported by the Swiss National Science Foundation within the collaborative DACH project Arrangements and Drawings as SNSF Project 200021E-171681. K. K. is supported by DFG Project MU 3501/3-1 and within the Research Training Group GRK 2434 Facets of Complexity. F. S. is supported by the German Research Foundation DFG Project FE 340/12-1.

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Notes

  1. 1.

    In the literature, usually more obstructions are mentioned, which we exclude for all drawings (simple or not), see Sect. 2.

  2. 2.

    In [17], these graphs are called fan-planar. We do not use this terminology to avoid mix-ups with the class of graphs admitting (not necessarily simple) fan-planar drawings.

  3. 3.

    More specifically, the statement and proof of [17, Lemma 1] are incorrect. A counterexample can be obtained by removing the edge g from the construction illustrated in Fig. 8 (vertices RB correspond to the vertices uw in [17, Lemma 1]); for a formal description of the construction see Lemma 3.

    After our submission to GD’21, the authors of [17] have uploaded a new version [18] of their preprint in which they state a different definition of fan-planarity with an additional forbidden crossing configuration; also see [18, last paragraph of Sect. 1].

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Correspondence to Meghana M. Reddy .

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Klemz, B., Knorr, K., Reddy, M.M., Schröder, F. (2021). Simplifying Non-simple Fan-Planar Drawings. In: Purchase, H.C., Rutter, I. (eds) Graph Drawing and Network Visualization. GD 2021. Lecture Notes in Computer Science(), vol 12868. Springer, Cham. https://doi.org/10.1007/978-3-030-92931-2_4

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  • DOI: https://doi.org/10.1007/978-3-030-92931-2_4

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