On fan-crossing and fan-crossing free graphs

Brandenburg, Franz J.

doi:10.1016/j.ipl.2018.06.006

Cited by 8 publications

(3 citation statements)

References 51 publications

(106 reference statements)

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“…A graph G is optimal 2-planar if and only if the planar skeleton is a 3-connected pentangulation [11], so that G is obtained by adding a pentagram of crossed edges in each pentagon. Every optimal 2-planar graph is an optimal fancrossing [15], fan-planar [38], 1-gap planar [10], and 5-map graph [18]. There are optimal 2-planar graphs for every n ≥ 50 with n = 2 mod 3 [42].…”

Section: Preliminariesmentioning

confidence: 99%

On Optimal Beyond-Planar Graphs

Brandenburg¹

2022

Preprint

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A graph is beyond-planar if it can be drawn in the plane with a specific restriction on crossings. Several types of beyond-planar graphs have been investigated, such as k-planar if every edge is crossed at most k times and RAC if edges can cross only at a right angle in a straight-line drawing. A graph is optimal if the number of edges coincides with the density for its type. Optimal graphs are special and are known only for some types of beyond-planar graphs, including 1-planar, 2-planar, and RAC graphs. For all types of beyond-planar graphs for which optimal graphs are known, we compute the range for optimal graphs, establish combinatorial properties, and show that every graph is a topological minor of an optimal graph. Note that the minor property is well-known for general beyond-planar graphs.

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Section: Preliminariesmentioning

confidence: 99%

On Optimal Beyond-Planar Graphs

Brandenburg¹

2022

Preprint

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“…Then only crossings by independent edges are allowed, that is the edges have distinct vertices. Note that there are graphs that are fan-crossing and fan-crossing free, but not 1-planar [23]. Straightline fan-crossing drawings are fan-planar [24], which are fan-crossing and exclude crossings of an edge from both sides.…”

Section: Introductionmentioning

confidence: 99%

Straight-line Drawings of 1-Planar Graphs

Brandenburg¹

2021

Preprint

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A graph is 1-planar if it can be drawn in the plane so that each edge is crossed at most once. However, there are 1-planar graphs which do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a two-coloring of the edges, so that edges of the same color do not cross. Hence, 1-planar graphs have geometric thickness two. In addition, each edge is crossed by edges with a common vertex if it is crossed more than twice. The drawings use high precision arithmetic with numbers with O(n log n) digits and can be computed in linear time from a 1-planar drawing.

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“…More recently, several other classes have been suggested (e.g., [6,11]), also motivated by cognitive experiments [32,37] indicating that the absence of certain types of crossings helps in improving the readability of a drawing; for a survey, refer to [24]. Some of the most studied are: (i) fan-planar graphs, in which no edge can be crossed by two independent edges or by two adjacent edges from different directions [12,13,33], (ii) fan-crossing free graphs, in which no edge can be crossed by two adjacent edges [16,19], (iii) gap-planar graphs, in which each crossing is assigned to one of its two involved edges, such that each edge can be assigned at most one crossing [11], and (iv) RAC graphs, in which edge crossings occur only at right angles [22,23,25]; see Figs. 1c-1e.…”

Section: Introductionmentioning

confidence: 99%

Efficient Generation of Different Topological Representations of Graphs Beyond-Planarity

Angelini¹,

Bekos²,

Kaufmann³

et al. 2019

Preprint

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Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph class for its maximum edge density, another parameter that is often considered in the literature is the size of the largest complete or complete bipartite graph belonging to it. Overcoming the limitations of standard combinatorial arguments, we present a technique to systematically generate all non-isomorphic topological representations of complete and complete bipartite graphs, taking into account the constraints of the specific class. As a proof of concept, we apply our technique to various beyond-planarity classes and achieve new tight bounds for the aforementioned parameter.

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On fan-crossing and fan-crossing free graphs

Cited by 8 publications

References 51 publications

On Optimal Beyond-Planar Graphs

On Optimal Beyond-Planar Graphs

Straight-line Drawings of 1-Planar Graphs

Efficient Generation of Different Topological Representations of Graphs Beyond-Planarity

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