Crossing Numbers of Beyond-Planar Graphs

Chimani, Markus; Kindermann, Philipp; Montecchiani, Fabrizio; Valtr, Pável

doi:10.1007/978-3-030-35802-0_6

Cited by 4 publications

(15 citation statements)

References 23 publications

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“…In fact, the first preliminary version of this paper dates back to 2014 [34]. Since then, fan-planarity has become a very popular subject of study among several researchers [3,7,10,[12][13][14][15][16][17][18]22,23,33]. The interested reader may also have a look at the recent surveys on fan-planarity [11] or general beyond-planar graphs [26].…”

Section: Introductionmentioning

confidence: 99%

The Density of Fan-Planar Graphs

Kaufmann

Ueckerdt²

2022

Electron. J. Combin.

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A topological drawing of a graph is fan-planar if for each edge $e$ the edges crossing $e$ form a star and no endpoint of $e$ is enclosed by $e$ and its crossing edges. A fan-planar graph is a graph admitting such a drawing. Equivalently, this can be formulated by three forbidden patterns, one of which is the configuration where $e$ is crossed by two independent edges and the other two where $e$ is crossed by two incident edges in a way that encloses some endpoint of $e$. A topological drawing is simple if any two edges have at most one point in common. Fan-planar graphs are a new member in the ever-growing list of topological graphs defined by forbidden intersection patterns, such as planar graphs and their generalizations, Turán-graphs and Conway's thrackle conjecture. Hence fan-planar graphs fall into an important field in combinatorial geometry with applications in various areas of discrete mathematics. As every $1$-planar graph is fan-planar and every fan-planar graph is $3$-quasiplanar, they also fit perfectly in a recent series of works on nearly-planar graphs from the areas of graph drawing and combinatorial embeddings. In this paper we show that every fan-planar graph on $n$ vertices has at most $5n-10$ edges, even though a fan-planar drawing may have a quadratic number of crossings. Our bound, which is tight for every $n \geq 20$, indicates how nicely fan-planar graphs fit in the row with planar graphs ($3n-6$ edges) and $1$-planar graphs ($4n-8$ edges). With this, fan-planar graphs form an inclusion-wise largest non-trivial class of topological graphs defined by forbidden patterns, for which the maximum number of edges on $n$ vertices is known exactly. We demonstrate that maximum fan-planar graphs carry a rich structure, which makes this class attractive for many algorithms commonly used in graph drawing. Finally, we discuss possible extensions and generalizations of these new concepts.

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Section: Introductionmentioning

confidence: 99%

The Density of Fan-Planar Graphs

Kaufmann

Ueckerdt²

2022

Electron. J. Combin.

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“…Fan-planar Two independent edges crossing a third or two adjacent edges crossing another edge from different "sides" Ω(n) [9] O(n 2 ) [9] (k, l)-grid-free…”

Section: Familymentioning

confidence: 99%

“…In a k-planar drawing no edge can be involved in more than k crossings. Chimani et al [9] prove a tight bound on the crossing ratio for 1-planarity:…”

Section: K-planar Graphsmentioning

confidence: 99%

“…Proof. We construct the graph G as described by Chimani et al [9], Section 2 (see Figure 2). The construction of G consists of three parts: a rigid graph P (round vertices, black edges); its dual P * (square vertices, purple); a set of binding edges which connects P to P * (grey); and one special edge (orange).…”

Section: K-planar Graphsmentioning

confidence: 99%

“…Results. Chimani et al [9] gave bounds on the crossing ratio for the 1-planar, quasi-planar, and fan-planar families. These results show that there exist graphs that have significantly larger crossing numbers when drawn with the beyond-planar restrictions.…”

Section: Introductionmentioning

confidence: 99%

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Crossing Numbers of Beyond-Planar Graphs Revisited

Beusekom¹,

Parada²,

Speckmann³

2021

Preprint

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Graph drawing beyond planarity focuses on drawings of high visual quality for non-planar graphs which are characterized by certain forbidden edge configurations. A natural criterion for the quality of a drawing is the number of edge crossings. The question then arises whether beyond-planar drawings have a significantly larger crossing number than unrestricted drawings. Chimani et al. [GD'19] gave bounds for the ratio between the crossing number of three classes of beyond-planar graphs and the unrestricted crossing number. In this paper we extend their results to the main currently known classes of beyond-planar graphs characterized by forbidden edge configurations and answer several of their open questions.

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