Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends

Steven Chaplick; Fabian Lipp; Alexander Wolff; Johannes Zink

doi:10.1007/978-3-030-04414-5_10

Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends

Steven Chaplick
, Fabian Lipp
, Alexander Wolff
, Johannes Zink^*

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

Abstract

We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two pairs of crossing edges share two vertices. We study the relations of these beyond-planar graph classes to right-angle crossing (RAC) graphs that admit compact drawings on the grid with few bends. We present four drawing algorithms that preserve the given embeddings. First, we show that every n-vertex NIC-planar graph admits a NIC-planar RAC drawing with at most one bend per edge on a grid of size O(n)×O(n). Then, we show that every n-vertex 1-planar graph admits a 1-planar RAC drawing with at most two bends per edge on a grid of size O(n ³)×O(n ³). Finally, we make two known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at most one bend per edge and for drawing IC-planar RAC graphs straight-line.

Original language	English
Title of host publication	Graph Drawing and Network Visualization - 26th International Symposium, GD 2018, Proceedings
Editors	Therese Biedl, Andreas Kerren
Publisher	Springer, Cham
Pages	137-151
Number of pages	15
ISBN (Electronic)	9783030044145
ISBN (Print)	9783030044138
DOIs	https://doi.org/10.1007/978-3-030-04414-5_10
Publication status	Published - 2018
Externally published	Yes

Publication series

Series	Lecture Notes in Computer Science
Volume	11282
ISSN	0302-9743

Access to Document

10.1007/978-3-030-04414-5_10

1 Article

Compact drawings of 1-planar graphs with right-angle crossings and few bends
Chaplick, S., Lipp, F., Wolff, A. & Zink, J., Nov 2019, In: Computational Geometry. 84, p. 50-68 19 p.
Research output: Contribution to journal › Article › Academic › peer-review

Open Access

Cite this

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title = "Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends",

abstract = "We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two pairs of crossing edges share two vertices. We study the relations of these beyond-planar graph classes to right-angle crossing (RAC) graphs that admit compact drawings on the grid with few bends. We present four drawing algorithms that preserve the given embeddings. First, we show that every n-vertex NIC-planar graph admits a NIC-planar RAC drawing with at most one bend per edge on a grid of size O(n){\texttimes}O(n). Then, we show that every n-vertex 1-planar graph admits a 1-planar RAC drawing with at most two bends per edge on a grid of size O(n 3){\texttimes}O(n 3). Finally, we make two known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at most one bend per edge and for drawing IC-planar RAC graphs straight-line.",

author = "Steven Chaplick and Fabian Lipp and Alexander Wolff and Johannes Zink",

note = "DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.",

year = "2018",

doi = "10.1007/978-3-030-04414-5\_10",

language = "English",

isbn = "9783030044138",

series = "Lecture Notes in Computer Science",

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Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends. / Chaplick, Steven; Lipp, Fabian; Wolff, Alexander et al.
Graph Drawing and Network Visualization - 26th International Symposium, GD 2018, Proceedings. ed. / Therese Biedl; Andreas Kerren. Springer, Cham, 2018. p. 137-151 (Lecture Notes in Computer Science, Vol. 11282).

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

TY - GEN

T1 - Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends

AU - Chaplick, Steven

AU - Lipp, Fabian

AU - Wolff, Alexander

AU - Zink, Johannes

N1 - DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

PY - 2018

Y1 - 2018

N2 - We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two pairs of crossing edges share two vertices. We study the relations of these beyond-planar graph classes to right-angle crossing (RAC) graphs that admit compact drawings on the grid with few bends. We present four drawing algorithms that preserve the given embeddings. First, we show that every n-vertex NIC-planar graph admits a NIC-planar RAC drawing with at most one bend per edge on a grid of size O(n)×O(n). Then, we show that every n-vertex 1-planar graph admits a 1-planar RAC drawing with at most two bends per edge on a grid of size O(n 3)×O(n 3). Finally, we make two known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at most one bend per edge and for drawing IC-planar RAC graphs straight-line.

AB - We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two pairs of crossing edges share two vertices. We study the relations of these beyond-planar graph classes to right-angle crossing (RAC) graphs that admit compact drawings on the grid with few bends. We present four drawing algorithms that preserve the given embeddings. First, we show that every n-vertex NIC-planar graph admits a NIC-planar RAC drawing with at most one bend per edge on a grid of size O(n)×O(n). Then, we show that every n-vertex 1-planar graph admits a 1-planar RAC drawing with at most two bends per edge on a grid of size O(n 3)×O(n 3). Finally, we make two known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at most one bend per edge and for drawing IC-planar RAC graphs straight-line.

U2 - 10.1007/978-3-030-04414-5_10

DO - 10.1007/978-3-030-04414-5_10

M3 - Conference article in proceeding

SN - 9783030044138

T3 - Lecture Notes in Computer Science

SP - 137

EP - 151

BT - Graph Drawing and Network Visualization - 26th International Symposium, GD 2018, Proceedings

A2 - Biedl, Therese

A2 - Kerren, Andreas

PB - Springer, Cham

ER -

Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends

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Compact drawings of 1-planar graphs with right-angle crossings and few bends

Cite this