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Embedding-Preserving Rectangle Visibility Representations of Nonplanar Graphs

  • Therese Biedl
  • Giuseppe Liotta
  • Fabrizio Montecchiani
Article

Abstract

A (weak) rectangle visibility representation, or simply an 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. Given a graph with a fixed embedding in the plane, we show that the problem of testing whether this graph has an embedding-preserving RVR can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs, i.e., for embedded graphs having at most one crossing per edge. The linear time algorithm uses three forbidden configurations, which extend the set known for straight-line drawings of 1-plane graphs. The algorithm first checks for the presence of these forbidden configurations in the input graph, and then either an embedding-preserving RVR is computed (also in linear time) or a forbidden configuration is reported as a negative witness. Finally, we discuss extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.

Keywords

Visibility representations 1-Planarity Fixed embedding Forbidden configuration 

Mathematics Subject Classification

68U05 68R10 94C15 

Notes

Acknowledgements

We thank the anonymous referees of this paper for their valuable suggestions.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Dipartimento di IngegneriaUniversità degli Studi di PerugiaPerugiaItaly

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