Testing Simultaneous Planarity when the Common Graph is 2-Connected

Haeupler, Bernhard; Jampani, Krishnam Raju; Lubiw, Anna

doi:10.7155/jgaa.00289

Cited by 27 publications

(28 citation statements)

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“…The complexity of simultaneous planarity for two graphs remains open, but it is tempting to conjecture that it is in P. In fact, we will state a (combinatorial) conjecture later which implies that simultaneous planarity is in P. Several special cases have been solved recently. One can test whether (G 1 , G 2 ) is simultaneously planar (i) in linear time if G 1 ∩G 2 is 2-connected (Haeupler, Jampani, Lubiw [22,23]), (ii) in linear time if G 1 ∩ G 2 consists of disjoint cycles (Bläsius, Rutter [9]), (iii) in quadratic time if a fixed embedding of each connected component of G 1 ∩ G 2 is given (Bläsius, Rutter [9]), (iv) in quadratic time if G 1 and G 2 are 2-connected, and G 1 ∩ G 2 is connected (Bläsius,Rutter [8]).…”

Section: Simultaneous Graph Drawingmentioning

confidence: 99%

Toward a Theory of Planarity: Hanani-Tutte and Planarity Variants

Schaefer¹

2013

JGAA

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We study Hanani-Tutte style theorems for various notions of planarity, including partially embedded planarity, and simultaneous planarity. This approach brings together the combinatorial, computational and algebraic aspects of planarity notions and may serve as a uniform foundation for planarity, as suggested in the writings of Tutte and Wu. 1 There were precursors to his approach, notably the paper by Hanani [12], but also work by Flores, van Kampen, and Wu. Some of the history can be found in [33].

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Section: Simultaneous Graph Drawingmentioning

confidence: 99%

Toward a Theory of Planarity: Hanani-Tutte and Planarity Variants

Schaefer¹

2013

JGAA

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“…1a is empty, or has a sufficiently simple structure, G 1 , G 2 , G 3 admits a QuaSEFE. Most notably, this is always the case in the sunflower setting [4,21,24], in which every edge belongs either to a single graph or to all graphs, i.e., H 1,2 = H 1,3 = H 2,3 = ∅. We extend this result to any set of planar graphs.…”

Section: Sufficient Conditions For Quasefesmentioning

confidence: 75%

The QuaSEFE Problem

Angelini

Förster

Hoffmann

et al. 2019

Lecture Notes in Computer Science

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We initiate the study of Simultaneous Graph Embedding with Fixed Edges in the beyond planarity framework. In the QuaSEFE problem, we allow edge crossings, as long as each graph individually is drawn quasiplanar, that is, no three edges pairwise cross. We show that a triple consisting of two planar graphs and a tree admit a QuaSEFE. This result also implies that a pair consisting of a 1-planar graph and a planar graph admits a QuaSEFE. We show several other positive results for triples of planar graphs, in which certain structural properties for their common subgraphs are fulfilled. For the case in which simplicity is also required, we give a triple consisting of two quasiplanar graphs and a star that does not admit a QuaSEFE. Moreover, in contrast to the planar SEFE problem, we show that it is not always possible to obtain a QuaSEFE for two matchings if the quasiplanar drawing of one matching is fixed.

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“…Haeupler et al [17] showed that if two simultaneously planar graphs G 1 and G 2 share a subgraph G that is connected, then there is a simultaneous planar drawing in which any edge of G 1 and any edge of G 2 intersect at most once. Introducing vertices at crossing points yields a planar graph, and a straight-line drawing of that graph provides a simultaneous planar drawing with O(n) bends per edge, O(n) crossings per edge, and with vertices, bends, and crossings on an O(n 2 ) × O(n 2 ) grid.…”

Section: Related Workmentioning

confidence: 99%

“…This will involve introducing new vertices where edges cross into the disc. The same idea was used in [17,Theorem 2].…”

Section: Extending Partial Embeddingsmentioning

confidence: 99%

Drawing Partially Embedded and Simultaneously Planar Graphs

Chan¹,

Frati²,

Gutwenger³

et al. 2015

JGAA

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Abstract. We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem-to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph-and the simultaneous planarity problem-to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required planar drawings exist, then there are planar drawings with a linear number of bends per edge and, in the case of simultaneous planarity, a constant number of crossings between every pair of edges. Our proofs provide efficient algorithms if the combinatorial embedding of the drawing is given. Our result on partially embedded graph drawing generalizes a classic result by Pach and Wenger which shows that any planar graph can be drawn with a linear number of bends per edge if the location of each vertex is fixed.

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Testing Simultaneous Planarity when the Common Graph is 2-Connected

Cited by 27 publications

References 0 publications

Toward a Theory of Planarity: Hanani-Tutte and Planarity Variants

Toward a Theory of Planarity: Hanani-Tutte and Planarity Variants

The QuaSEFE Problem

Drawing Partially Embedded and Simultaneously Planar Graphs

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