Recognizing and drawing IC-planar graphs

Brandenburg, Franz J.; Didimo, Walter; Evans, William S.; Kindermann, Philipp; Liotta, Giuseppe; Montecchiani, Fabrizio

doi:10.1016/j.tcs.2016.04.026

Cited by 51 publications

(63 citation statements)

References 54 publications

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“…The 1-planarity problem remains NP-complete even for graphs with bounded bandwidth, pathwidth, or treewidth [12], and for graphs obtained from planar graphs by adding a single edge [34]. Brandenburg et al [29] proved that the problem of deciding whether a graph G is IC-planar is also NP-complete, by using a reduction from the 1-planarity problem. A similar reduction holds for NIC-planar graphs as well [10].…”

Section: Np-hardness Resultsmentioning

confidence: 99%

An annotated bibliography on 1-planarity

Kobourov

Liotta

Montecchiani

2017

Computer Science Review

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115

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The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back to more than fifty years ago and, recently, it has driven increasing attention in the areas of graph theory, graph algorithms, graph drawing, and computational geometry. This annotated bibliography aims to provide a guiding reference to researchers who want to have an overview of the large body of literature about 1-planar graphs. It reviews the current literature covering various research streams about 1-planarity, such as characterization and recognition, combinatorial properties, and geometric representations. As an additional contribution, we offer a list of open problems on 1-planar graphs.

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Section: Np-hardness Resultsmentioning

confidence: 99%

An annotated bibliography on 1-planarity

Kobourov

Liotta

Montecchiani

2017

Computer Science Review

Self Cite

115

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“…• Deciding whether a graph is 1-gap-planar is NP-complete, even when the drawing of a given graph is restricted to a fixed rotation system that is part of the input (Section 6). Note that analogous recognition problems for other families of beyond-planar graphs are also NP-hard (see, e.g., [7,10,14,15,31,43]), while polynomial algorithms are known in some restricted settings (see, e.g., [6,10,15,20,24,37,35]).…”

Section: Introductionmentioning

confidence: 99%

Gap-planar graphs

Bae

Baffier

Jin

et al. 2018

Theoretical Computer Science

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We introduce the family of k-gap-planar graphs for k ≥ 0, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most k of its crossings. This definition is motivated by applications in edge casing, as a k-gap-planar graph can be drawn crossing-free after introducing at most k local gaps per edge. We present results on the maximum density of k-gap-planar graphs, their relationship to other classes of beyond-planar graphs, characterization of k-gap-planar complete graphs, and the computational complexity of recognizing kgap-planar graphs.

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“…The second part follows from the fact that the (4, 1)-planar graphs coincide with the IC-planar graphs [13]. Testing IC-planarity is known to be NP-complete [4]. Appendix B proves both equivalencies.…”

Section: Recognition Of (K P)-planar Graphsmentioning

confidence: 90%

(k, p)-Planarity: A Relaxation of Hybrid Planarity

Giacomo

Lenhart

Liotta

et al. 2018

WALCOM: Algorithms and Computation

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We present a new model for hybrid planarity that relaxes existing hybrid representations. A graph G = (V, E) is (k, p)-planar if V can be partitioned into clusters of size at most k such that G admits a drawing where: (i) each cluster is associated with a closed, bounded planar region, called a cluster region; (ii) cluster regions are pairwise disjoint, (iii) each vertex v ∈ V is identified with at most p distinct points, called ports, on the boundary of its cluster region; (iv) each inter-cluster edge (u, v) ∈ E is identified with a Jordan arc connecting a port of u to a port of v; (v) inter-cluster edges do not cross or intersect cluster regions except at their endpoints. We first tightly bound the number of edges in a (k, p)-planar graph with p < k. We then prove that (4, 1)-planarity testing and (2, 2)-planarity testing are NP-complete problems. Finally, we prove that neither the class of (2, 2)-planar graphs nor the class of 1planar graphs contains the other, indicating that the (k, p)-planar graphs are a large and novel class.

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Recognizing and drawing IC-planar graphs

Cited by 51 publications

References 54 publications

An annotated bibliography on 1-planarity

An annotated bibliography on 1-planarity

Gap-planar graphs

(k, p)-Planarity: A Relaxation of Hybrid Planarity

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