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: Contribution to journalArticleAcademicpeer-review

2 Citations (Web of Science)

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 (beyond-planar graphs is a collective term for the primary attempts to generalize the planar graphs) 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) x 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)) x 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. (C) 2019 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)50-68
Number of pages19
JournalComputational Geometry
Volume84
DOIs
Publication statusPublished - Nov 2019
Externally publishedYes

Keywords

  • 1-planar graphs
  • ALGORITHM
  • Beyond-planar graphs
  • Graph drawing
  • Grid drawing
  • PLANAR GRAPH
  • Right-angle crossings

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