Simultaneous Embedding of Planar Graphs with Few Bends

Erten, Cesim; Kobourov, Stephen G.

doi:10.1007/978-3-540-31843-9_21

Cited by 35 publications

(64 citation statements)

References 17 publications

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“…Proof. Our algorithm is very similar to algorithms due to Brass et al [6] and Erten and Kobourov [10]. These algorithms, however, do not guarantee the construction of a simultaneous embedding in which the order of the edges incident to each vertex is as stated in the lemma.…”

Section: Lemmamentioning

confidence: 98%

“…In Step 3, we independently augment R ′ and B ′ to Hamiltonian planar graphs, so as to satisfy topological constraints that are necessary for the subsequent drawing algorithms. In Step 4, we use the Hamiltonian augmentations to construct a simultaneous embedding of R ′ and B ′ with one bend per edge; this step is reminiscent of an algorithm of Erten and Kobourov [10]. Finally, in Step 5, we expand the components of C. This consists of modifying the simultaneous embedding of R ′ and B ′ in a small neighborhood of each vertex to make room for the components of C. We now describe these steps in detail.…”

Section: Two Treesmentioning

confidence: 99%

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Simultaneous Embeddings with Few Bends and Crossings

Frati

Hoffmann

Kusters

2015

Lecture Notes in Computer Science

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A simultaneous embedding with fixed edges (SEFE) of two planar graphs R and B is a pair of plane drawings of R and B that coincide when restricted to the common vertices and edges of R and B. We show that whenever R and B admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve with few bends and every pair of edges has few crossings. Specifically:(1) if R and B are trees then one bend per edge and four crossings per edge pair suffice (and one bend per edge is sometimes necessary), (2) if R is a planar graph and B is a tree then six bends per edge and eight crossings per edge pair suffice, and (3) if R and B are planar graphs then six bends per edge and sixteen crossings per edge pair suffice. Our results improve on a paper by Grilli et al. (GD'14), which proves that nine bends per edge suffice, and on a paper by Chan et al. (GD'14), which proves that twenty-four crossings per edge pair suffice.

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Section: Lemmamentioning

confidence: 98%

Section: Two Treesmentioning

confidence: 99%

Simultaneous Embeddings with Few Bends and Crossings

Frati

Hoffmann

Kusters

2015

Lecture Notes in Computer Science

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“…A simultaneous embedding Γ of G 1 and G 2 consists of their planar drawings D 1 and D 2 , where each vertex is mapped to the same point in the plane in both D 1 and D 2 . Erten and Kobourov [14] showed that every pair of planar graphs admit a simultaneous embedding with at most three bends per edge. Giacomo and Liotta [16] observed that by using monotone topological book embeddings Erten and Kobourov's [14] construction can achieve a drawing with two bends per edge.…”

Section: Technical Detailsmentioning

confidence: 99%

Relating Graph Thickness to Planar Layers and Bend Complexity

Durocher¹,

Mondal²

2018

SIAM J. Discrete Math.

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The thickness of a graph G = (V, E) with n vertices is the minimum number of planar subgraphs of G whose union is G. A polyline drawing of G in R 2 is a drawing Γ of G, where each vertex is mapped to a point and each edge is mapped to a polygonal chain. Bend and layer complexities are two important aesthetics of such a drawing. The bend complexity of Γ is the maximum number of bends per edge in Γ, and the layer complexity of Γ is the minimum integer r such that the set of polygonal chains in Γ can be partitioned into r disjoint sets, where each set corresponds to a planar polyline drawing. Let G be a graph of thickness t. By Fáry's theorem, if t = 1, then G can be drawn on a single layer with bend complexity 0. A few extensions to higher thickness are known, e.g., if t = 2 (resp., t > 2), then G can be drawn on t layers with bend complexity 2 (resp., 3n + O(1)). However, allowing a higher number of layers may reduce the bend complexity, e.g., complete graphs require Θ(n) layers to be drawn using 0 bends per edge.In this paper we present an elegant extension of Fáry's theorem to draw graphs of thickness t > 2. We first prove that thickness-t graphs can be drawn on t layers with 2.25n+O(1) bends per edge. We then develop another technique to draw thickness-t graphs on t layers with bend complexity, i.e., O( √ 2 t · n 1−(1/β) ), where β = 2 (t−2)/2 . Previously, the bend complexity was not known to be sublinear for t > 2. Finally, we show that graphs with linear arboricity k can be drawn on k layers with bend complexity 3(k−1)n (4k−2) .

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“…The problem of computing a simultaneous embedding of two or more graphs has been extensively explored by the graph drawing community. Indeed, besides its inherent theoretical interest [1,2,4,5,6,7,9,10,11,12,13,14,15,16,17,18,19,22,23,24,25,26], it has several applications in dynamic network visualization, especially when a visual analysis of an evolving network is needed. Although many variants of this problem have been investigated so far, a general formulation for two graphs can be stated as follows: Let G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) be two planar graphs sharing a common (or shared) subgraph G = (V, E), where V = V 1 ∩ V 2 and E = E 1 ∩ E 2 .…”

Section: Introductionmentioning

confidence: 99%

On the NP-hardness of GRacSim drawing and k-SEFE Problems

Grilli¹

2018

JGAA

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We study the complexity of two problems in simultaneous graph drawing. The first problem, GRACSIM DRAWING, asks for finding a simultaneous geometric embedding of two graphs such that only crossings at right angles are allowed. The second problem, K-SEFE, is a restricted version of the topological simultaneous embedding with fixed edges (SEFE) problem, for two planar graphs, in which every private edge may receive at most k crossings, where k is a prescribed positive integer. We show that GRACSIM DRAWING is N P-hard and that K-SEFE is N P-complete. The N P-hardness of both problems is proved using two similar reductions from 3-PARTITION.

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Simultaneous Embedding of Planar Graphs with Few Bends

Cited by 35 publications

References 17 publications

Simultaneous Embeddings with Few Bends and Crossings

Simultaneous Embeddings with Few Bends and Crossings

Relating Graph Thickness to Planar Layers and Bend Complexity

On the NP-hardness of GRacSim drawing and k-SEFE Problems

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