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

Schaefer, Marcus

doi:10.7155/jgaa.00298

Cited by 63 publications

(79 citation statements)

References 17 publications

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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%

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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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Section: Sufficient Conditions For Quasefesmentioning

confidence: 75%

“…We extend this result to any set of planar graphs. We remark that SEFE is NP-complete in the sunflower setting for three planar graphs [4,24].…”

Section: Sufficient Conditions For Quasefesmentioning

confidence: 97%

The QuaSEFE Problem

Angelini

Förster

Hoffmann

et al. 2019

Lecture Notes in Computer Science

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“…Finally, we give a short polynomial time reduction of c-planarity to thickenability. By combinining the results of Schaefer [40,Theorem 6.17], and Angelini and Da Lozzo [4] we observe that c-planarity is polynomial-time equivalent to connected sefe-2, the problem of deciding simultaneous embeddability of two graphs in the case when the intersection of the two graphs is connected (see Section 4 for formal statement of the problem). The general version of the problem, known as sefe-2, where the intersection of the two graphs may be disconnected, is notoriously difficult.…”

Section: Introductionmentioning

confidence: 78%

Atomic Embeddability, Clustered Planarity, and Thickenability

Fulek

Tóth

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the atomic embeddability testing problem, which is a common generalization of clustered planarity (c-planarity, for short) and thickenability testing, and present a polynomial time algorithm for this problem, thereby giving the first polynomial time algorithm for c-planarity.C-planarity was introduced in 1995 by Feng, Cohen, and Eades as a variant of graph planarity, in which the vertex set of the input graph is endowed with a hierarchical clustering and we seek an embedding (crossing free drawing) of the graph in the plane that respects the clustering in a certain natural sense. Until now, it has been an open problem whether c-planarity can be tested efficiently, despite relentless efforts. The thickenability problem for simplicial complexes emerged in the topology of manifolds in the 1960s. A 2-dimensional simplicial complex is thickenable if it embeds in some orientable 3-dimensional manifold. Recently, Carmesin announced that thickenability can be tested in polynomial time.Our algorithm for atomic embeddability combines ideas from Carmesin's work with algorithmic tools previously developed for weak embeddability testing. We express our results purely in terms of graphs on surfaces, and rely on the machinery of topological graph theory.Finally we give a polynomial-time reduction from c-planarity to thickenability and show that a slight generalization of atomic embeddability to the setting in which clusters are toroidal graphs is NP-complete.

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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%

“…In contrast, it is a long-standing open problem to determine whether the existence of a SEFE can be tested in polynomial time or not, for two planar graphs; though, the testing problem is N P-complete when generalizing SEFE to three or more graphs [22]. However, several polynomial time testing algorithms have been provided under different assumptions [3,4,11,12,24,26], most of them involve the connectivity or the maximum degree of the input graphs or of their common subgraph.…”

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

Cited by 63 publications

References 17 publications

The QuaSEFE Problem

The QuaSEFE Problem

Atomic Embeddability, Clustered Planarity, and Thickenability

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

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