On Optimal 2- and 3-Planar Graphs

Bekos, Michael A.; Kaufmann, Michael; Raftopoulou, Chrysanthi N.

doi:10.4230/lipics.socg.2017.16

Cited by 19 publications

(16 citation statements)

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“…Straight-line is also a real restriction for fan-crossing [18], fan-crossing free [19,20], and 2-planar graphs [21], which each generalize 1-planar graphs. Fancrossing graphs admit drawings with the crossing of an edge by edges of a fan, that is the crossing edges are incident to a common vertex, whereas such crossings Supported by Deutsche Forschungsgemeinschaft (DFG) Br835/20-1 are excluded for fan-crossing free graphs [22].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

“…A fan-crossing free drawing of an n-vertex graph with 4n−8 edges is 1-planar [20], which cannot be drawn straight-line [16]. Finally, the crossed dodecahedron graph is 2-planar and fan-crossing, but it does not admit a straight-line 2-planar drawing, since it has a unique 2-planar embedding [21], which is shown in Fig. 2(a).…”

Section: Introductionmentioning

confidence: 99%

Straight-line Drawings of 1-Planar Graphs

Brandenburg¹

2021

Preprint

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“…Binucci et al [9] have shown that for each k ≥ 2 the class of graphs admitting simple k-planar drawings and the class of graphs admitting simple fan-planar drawings are incomparable. In contrast, every so-called optimal 2-planar graph admits a simple fan-planar drawing [7]. This follows from the fact that these graphs can be characterized as the graphs obtained by drawing a pentagram in the interior of each face of a pentagulation [7], which yields a fan-planar drawing.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast, every so-called optimal 2-planar graph admits a simple fan-planar drawing [7]. This follows from the fact that these graphs can be characterized as the graphs obtained by drawing a pentagram in the interior of each face of a pentagulation [7], which yields a fan-planar drawing. Angelini et al [3] introduced a drawing style that combines fan-planarity with a visualization technique called edge bundling [14,15,20].…”

Section: Introductionmentioning

confidence: 99%

“…We spoke with the authors and they confirmed that the current version of their proof is not correct and that they do not see a simple way to fix it 7 . Kaufmann and Ueckerdt [17] described an infinite family of simple fan-planar drawings 7 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 Figure 8 (vertices R, B correspond to the vertices u, w in [17, Lemma 1]); for a formal description of the construction see Lemma 3.…”

Section: Introductionmentioning

confidence: 99%

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Simplifying Non-Simple Fan-Planar Drawings

Klemz¹,

Knorr²,

Reddy³

et al. 2021

Preprint

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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.

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Edge‐minimum saturated k‐planar drawings

Chaplick,

Klute,

Parada

et al. 2024

Journal of Graph Theory

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For a class of drawings of loopless (multi‐)graphs in the plane, a drawing is saturated when the addition of any edge to results in —this is analogous to saturated graphs in a graph class as introduced by Turán and Erdős, Hajnal, and Moon. We focus on ‐planar drawings, that is, graphs drawn in the plane where each edge is crossed at most times, and the classes of all ‐planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing more than once, or no edge crossing itself. While saturated ‐planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. We establish a generic framework to determine the minimum number of edges among all ‐vertex saturated ‐planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest ‐vertex saturated ‐planar drawings have edges for any , while if all that is forbidden, the sparsest such drawings have edges for any .

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On Optimal 2- and 3-Planar Graphs

Cited by 19 publications

References 0 publications

Straight-line Drawings of 1-Planar Graphs

Straight-line Drawings of 1-Planar Graphs

Simplifying Non-Simple Fan-Planar Drawings

Edge‐minimum saturated k‐planar drawings

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