License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2016.19
URN: urn:nbn:de:0030-drops-59116
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5911/
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Biedl, Therese ; Liotta, Giuseppe ; Montecchiani, Fabrizio

On Visibility Representations of Non-Planar Graphs

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LIPIcs-SoCG-2016-19.pdf (0.6 MB)

Abstract

A rectangle visibility representation (RVR) of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Testing whether a graph has an RVR is known to be NP-hard. In this paper, we study the problem of finding an RVR under the assumption that an embedding in the plane of the input graph is fixed and we are looking for an RVR that reflects this embedding. We show that in this case the problem can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs (i.e., embedded graphs having at most one crossing per edge). The linear time algorithm uses a precise list of forbidden configurations, which extends the set known for straight-line drawings of 1-plane graphs. These forbidden configurations can be tested for in linear time, and so in linear time we can test whether a 1-plane graph has an RVR and either compute such a representation or report a negative witness. Finally, we discuss some extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.

BibTeX - Entry

@InProceedings{biedl_et_al:LIPIcs:2016:5911,
  author =	{Therese Biedl and Giuseppe Liotta and Fabrizio Montecchiani},
  title =	{{On Visibility Representations of Non-Planar Graphs}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5911},
  URN =		{urn:nbn:de:0030-drops-59116},
  doi =		{10.4230/LIPIcs.SoCG.2016.19},
  annote =	{Keywords: Visibility Representations, 1-Planarity, Recognition Algorithm, Forbidden Configuration}
}

Keywords: Visibility Representations, 1-Planarity, Recognition Algorithm, Forbidden Configuration
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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