Fáry’s Theorem for 1-Planar Graphs

Hong, Seok‐Hee; Eades, Peter; Liotta, Giuseppe; Poon, Sheung-Hung

doi:10.1007/978-3-642-32241-9_29

Cited by 71 publications

(66 citation statements)

References 14 publications

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“…However, Thomassen [28] presented two forbidden subgraphs for straight-line drawings of 1-plane graphs. Hong et al [20] gave a linear-time testing and drawing algorithm to construct a straight-line 1-planar drawing, if it exists. Recently, Nagamochi solved the more general problem of straight-line drawability for wider classes of embedded graphs [23].…”

Section: Related Workmentioning

confidence: 99%

On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

et al. 2016

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Abstract. Fan-planar graphs were recently introduced as a generalization of 1-planar graphs. A graph is fan-planar if it can be embedded in the plane, such that each edge that is crossed more than once, is crossed by a bundle of two or more edges incident to a common vertex. A graph is outer-fan-planar if it has a fan-planar embedding in which every vertex is on the outer face. If, in addition, the insertion of an edge destroys its outer-fan-planarity, then it is maximal outer-fan-planar. In this paper, we present a polynomial-time algorithm to test whether a given graph is maximal outerfan-planar. The algorithm can also be employed to produce an outer-fan-planar embedding, if one exists. On the negative side, we show that testing fan-planarity of a graph is NP-hard, for the case where the rotation system (i.e., the cyclic order of the edges around each vertex) is given.

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Section: Related Workmentioning

confidence: 99%

On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

et al. 2016

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“…Thomassen [18] generalized Eggleton's result and characterized the class of 1-planar graphs which admit straight-line drawings by the exclusion of so-called B-and W-configurations in embeddings. These configurations were rediscovered by Hong et al [13], who also provide a linear-time drawing algorithm that starts from a given embedding.…”

Section: Who Calledmentioning

confidence: 99%

Recognizing Outer 1-Planar Graphs in Linear Time

et al. 2013

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Abstract.A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are on the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time and specialize 1-planar graphs, whose recognition is N P-hard.Our main result is a linear-time algorithm that first tests whether a graph G is o1p, and then computes an embedding. Moreover, the algorithm can augment G to a maximal o1p graph. If G is not o1p, then it includes one of six minors (see Fig. 3), which are also detected by the recognition algorithm. Hence, the algorithm returns a positive or negative witness for o1p.

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“…In graph drawing, 1-planarity has more recently become of interest, as a way of generalizing planar drawings in a controlled way that does not lead to too much visual complexity. Works in this area have compared 1-planarity to other forms of controlled crossings such as RAC (right-angle-crossing) graphs [11], found an algorithmic characterization of the 1-planar drawings that can be straightened to have all edges represented by straight line segments [17], and studied the transformation of rotation systems into 1-planar drawings [10]. However, until now there have been no published algorithms for finding 1-planar drawings of arbitrary graphs.…”

Section: Introductionmentioning

confidence: 99%

Parameterized Complexity of 1-Planarity

Bannister¹,

Cabello²,

Eppstein³

2018

JGAA

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We consider the problem of finding a 1-planar drawing for a general graph, where a 1-planar drawing is a drawing in which each edge participates in at most one crossing. Since this problem is known to be NP-hard we investigate the parameterized complexity of the problem with respect to the vertex cover number, tree-depth, and cyclomatic number. For these parameters we construct fixed-parameter tractable algorithms. However, the problem remains NP-complete for graphs of bounded bandwidth, pathwidth, or treewidth.

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Fáry’s Theorem for 1-Planar Graphs

Cited by 71 publications

References 14 publications

On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

Recognizing Outer 1-Planar Graphs in Linear Time

Parameterized Complexity of 1-Planarity

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