Algorithmica

, Volume 80, Issue 3, pp 977–994 | Cite as

On the Planar Split Thickness of Graphs

  • David Eppstein
  • Philipp Kindermann
  • Stephen Kobourov
  • Giuseppe Liotta
  • Anna Lubiw
  • Aude Maignan
  • Debajyoti Mondal
  • Hamideh Vosoughpour
  • Sue Whitesides
  • Stephen Wismath
Article
  • 58 Downloads
Part of the following topical collections:
  1. Special Issue on Theoretical Informatics

Abstract

Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest k such that the graph is k-splittable into a planar graph. A k-split operation substitutes a vertex v by at most k new vertices such that each neighbor of v is connected to at least one of the new vertices. We first examine the planar split thickness of complete graphs, complete bipartite graphs, multipartite graphs, bounded degree graphs, and genus-1 graphs. We then prove that it is NP-hard to recognize graphs that are 2-splittable into a planar graph, and show that one can approximate the planar split thickness of a graph within a constant factor. If the treewidth is bounded, then we can even verify k-splittability in linear time, for a constant k.

Keywords

Planarity Splittable Thickness Graph drawing Graph theory Complete graphs Genus-1 graphs NP-hardness Approximation Fixed-parameter tractable 

Notes

Acknowledgements

Most of the results of this paper were obtained at the McGill-INRIA-UVictoria Workshop on Computational Geometry, Barbados, February 2015. We would like to thank the organizers of these events, as well as many participants for fruitful discussions and suggestions. The first, fourth, sixth and eighth authors acknowledge the support from NSF Grant 1228639, 2012C4E3KT PRIN Italian National Research Project, PEPS egalite project and NSERC, respectively.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.University of CaliforniaIrvineUSA
  2. 2.FernUniversität HagenHagenGermany
  3. 3.University of ArizonaTucsonUSA
  4. 4.Università degli Studi di PerugiaPerugiaItaly
  5. 5.University of WaterlooWaterlooCanada
  6. 6.Universit. Grenoble AlpesGrenobleFrance
  7. 7.University of VictoriaVictoriaCanada
  8. 8.University of LethbridgeLethbridgeCanada

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