1-Planarity of Graphs with a Rotation System

Auer, Christopher; Brandenburg, Franz J.; Gleißner, Andreas; Reislhuber, Josef

doi:10.7155/jgaa.00347

Cited by 30 publications

(38 citation statements)

References 26 publications

(41 reference statements)

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“…A single K 4 implies that this property no longer holds for o1p graphs. In contrast, the recognition of 1-planar graphs is N P-hard [34], even if the graphs are given with a rotation system [6].…”

Section: Recognitionmentioning

confidence: 99%

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Outer 1-Planar Graphs

et al. 2015

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A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are in 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. We explore o1p graphs. Our first main result is a linear-time algorithm that takes a graph as input and returns a positive or a negative witness for o1p. If a graph G is o1p, then the algorithm computes an embedding and can augment G to a maximal o1p graph. Otherwise, G includes one of six minors, which is detected by the recognition algorithm. Secondly, we establish structural properties of o1p graphs. o1p graphs are planar and are subgraphs of planar graphs with a Hamiltonian cycle. They are neither closed under edge contraction nor This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) Grant Br835/18-1. B Christian Bachmaier

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Section: Recognitionmentioning

confidence: 99%

“…On the contrary, dealing with crossings generally leads to N P-hard problems. It is N P-hard to recognize 1-planar graphs [34], even if the graph is given with a rotation system, which determines the cyclic ordering of the edges at each vertex [6]. 1-planarity remains N P-hard even for bounded treewidth [7].…”

Section: Introductionmentioning

confidence: 99%

Outer 1-Planar Graphs

et al. 2015

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“…None (3,4,4,5,5,5,6) Fixed (3,4,5,5,5,6,6) Partial (4,4,5,5,6,6,6) Fixed (5,5,5,5,5,5,6) Unclear Infeasible…”

Section: Reduction Rules and Their Applicationmentioning

confidence: 98%

“…Again, H (x) is uniquely determined. 3 and v 1 is a red neighbor, which implies that v 1 is a black neighbor of x 1 and the case is decided. Hence, the coloring of a subgraph matching CC is fixed and its embedding is unique.…”

Section: If τ (X) = 5 Then the Edge Coloring Of H (X) Is Unclearmentioning

confidence: 99%

“…1-Planar embeddings are quite flexible, as the five embeddings of K 4 [29] and the N P-hardness proof of [3] show. There is an extension of Whitney's theorem by Schumacher [35] who proved that every 5-connected optimal 1-planar graph has a unique 1-planar embedding with the exception of the extended wheel graphs, which have two embeddings for graphs of size at least ten and eight for the minimum optimal 1-planar graph X W 6 with eight vertices.…”

Section: Introductionmentioning

confidence: 99%

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Recognizing Optimal 1-Planar Graphs in Linear Time

Brandenburg

2016

Algorithmica

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A graph with n vertices is 1-planar if it can be drawn in the plane such that each edge is crossed at most once, and is optimal if it has the maximum of 4n − 8 edges. We show that optimal 1-planar graphs can be recognized in linear time. Our algorithm implements a graph reduction system with two rules, which can be used to reduce every optimal 1-planar graph to an irreducible extended wheel graph. The graph reduction system is non-deterministic, constraint, and non-confluent.

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An Experimental Study of a 1-Planarity Testing and Embedding Algorithm

Binucci

Didimo

Montecchiani

2020

WALCOM: Algorithms and Computation

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The definition of 1-planar graphs naturally extends graph planarity, namely a graph is 1-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time, deciding whether a graph is 1-planar is NP-complete, even for restricted classes of graphs. Although several polynomial-time algorithms have been described for recognizing specific subfamilies of 1-planar graphs, no implementations of general algorithms are available to date. We investigate the feasibility of a 1-planarity testing and embedding algorithm based on a backtracking strategy. While the experiments show that our approach can be successfully applied to graphs with up to 30 vertices, they also suggest the need of more sophisticated techniques to attack larger graphs. Our contribution provides initial indications that may stimulate further research on the design of practical approaches for the 1-planarity testing problem.

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1-Planarity of Graphs with a Rotation System

Cited by 30 publications

References 26 publications

Outer 1-Planar Graphs

Outer 1-Planar Graphs

Recognizing Optimal 1-Planar Graphs in Linear Time

An Experimental Study of a 1-Planarity Testing and Embedding Algorithm

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