Lombardi Drawings of Graphs

Duncan, Christian A.; Eppstein, David; Goodrich, Michael T.; Kobourov, Stephen G.; Nöllenburg, Martin

doi:10.7155/jgaa.00251

Cited by 44 publications

(43 citation statements)

References 29 publications

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“…That is, we can find crossing-free circular arc drawings with perfect angular resolution and polynomial area. Recall that a drawing with circular arcs and perfect angular resolution is called a Lombardi drawing [11].…”

Section: Lombardi Drawings For Ordered Treesmentioning

confidence: 99%

“…In a separate paper [11] we study Lombardi drawings for classes of graphs other than trees. Unlike trees, not all planar graphs have planar Lombardi drawings [9,11] and it is an interesting open question to characterize the graphs that have a planar Lombardi drawing.…”

Section: Introductionmentioning

confidence: 99%

“…Unlike trees, not all planar graphs have planar Lombardi drawings [9,11] and it is an interesting open question to characterize the graphs that have a planar Lombardi drawing. Eppstein [14] recently proved that all planar subcubic graphs have a planar Lombardi drawing, and that there are 4-regular planar graphs that do not have a planar Lombardi drawing.…”

Section: Introductionmentioning

confidence: 99%

“…We define a Lombardi drawing [11] of a graph G as a drawing of G with perfect angular resolution such that each edge is drawn as a circular arc. When measuring the angle formed by two circular arcs incident to a node v, we use the angle formed by the tangents of the two arcs at v. Circular arcs are strictly more general than straight-line segments, since straight-line segments can be viewed as circular arcs with infinite radius.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Drawing Trees with Perfect Angular Resolution and Polynomial Area

Duncan

Eppstein

Goodrich

et al. 2012

Discrete Comput Geom

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We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2π/d(v). We show:1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area.2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution.3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area.Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.

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Section: Lombardi Drawings For Ordered Treesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Drawing Trees with Perfect Angular Resolution and Polynomial Area

Duncan

Eppstein

Goodrich

et al. 2012

Discrete Comput Geom

Self Cite

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“…Two features that stand out in Lombardi's work are the use of circular-arc edges and their even distribution around vertices. Such even spacing of edges around each vertex (also known as perfect angular resolution) together with the usage of circular arc edges, formally de-fine Lombardi drawings of graphs [6,7]. Chernobelskiy et al [5] relax the perfect angular resolution constraint in Lombardi drawings and describe functional force-directed algorithms, which produce aesthetically appealing near-Lombardi drawings.…”

Section: Related Workmentioning

confidence: 99%

Circle-Representations of Simple 4-Regular Planar Graphs

Bekos

Raftopoulou

2013

Graph Drawing

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Abstract. Lovász conjectured that every connected 4-regular planar graph G admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles, such that the vertices of G correspond to the intersection and touching points of the circles and the edges of G are the arc segments among pairs of intersection and touching points of the circles. In this paper, (a) we affirmatively answer Lovász's conjecture, if G is 3-connected, and, (b) we demonstrate an infinite class of connected 4-regular planar graphs which are not 3-connected and do not admit a realization as a system of circles.

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Orthogonal and Smooth Orthogonal Layouts of 1-Planar Graphs with Low Edge Complexity

Argyriou¹,

Cornelsen

Förster

et al. 2018

Lecture Notes in Computer Science

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While orthogonal drawings have a long history, smooth orthogonal drawings have been introduced only recently. So far, only planar drawings or drawings with an arbitrary number of crossings per edge have been studied. Recently, a lot of research effort in graph drawing has been directed towards the study of beyond-planar graphs such as 1-planar graphs, which admit a drawing where each edge is crossed at most once. In this paper, we consider graphs with a fixed embedding. For 1-planar graphs, we present algorithms that yield orthogonal drawings with optimal curve complexity and smooth orthogonal drawings with small curve complexity. For the subclass of outer-1-planar graphs, which can be drawn such that all vertices lie on the outer face, we achieve optimal curve complexity for both, orthogonal and smooth orthogonal drawings.

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Lombardi Drawings of Graphs

Cited by 44 publications

References 29 publications

Drawing Trees with Perfect Angular Resolution and Polynomial Area

Drawing Trees with Perfect Angular Resolution and Polynomial Area

Circle-Representations of Simple 4-Regular Planar Graphs

Orthogonal and Smooth Orthogonal Layouts of 1-Planar Graphs with Low Edge Complexity

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