On simultaneous planar graph embeddings

Peter Brass, Eowyn Cenek, Cristian A. Duncan, Alon Efrat, Cesim Erten, Dan P. Ismailescu, Stephen G Kobourov, Anna Lubiw, Joseph S B Mitchell

Research output: Contribution to journalArticle

84 Citations (Scopus)

Abstract

We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. We present positive and negative results for the two versions of the problem. Among the positive results with given mapping, we show that we can embed two paths on an n×n grid, and two caterpillar graphs on a 3n×3n grid. Among the negative results with given mapping, we show that it is not always possible to simultaneously embed three paths or two general planar graphs. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n)×O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O( n2)×O( n2) grid.

Original languageEnglish (US)
Pages (from-to)117-130
Number of pages14
JournalComputational Geometry: Theory and Applications
Volume36
Issue number2
DOIs
StatePublished - Feb 2007

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Graph Embedding
Planar graph
Grid
Caterpillar
Outerplanar Graph
Path
Graph in graph theory

Keywords

  • Graph drawing
  • Planar graphs
  • Simultaneous visualizations

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Computer Science Applications
  • Computational Mathematics
  • Control and Optimization
  • Geometry and Topology

Cite this

Brass, P., Cenek, E., Duncan, C. A., Efrat, A., Erten, C., Ismailescu, D. P., ... Mitchell, J. S. B. (2007). On simultaneous planar graph embeddings. Computational Geometry: Theory and Applications, 36(2), 117-130. https://doi.org/10.1016/j.comgeo.2006.05.006

On simultaneous planar graph embeddings. / Brass, Peter; Cenek, Eowyn; Duncan, Cristian A.; Efrat, Alon; Erten, Cesim; Ismailescu, Dan P.; Kobourov, Stephen G; Lubiw, Anna; Mitchell, Joseph S B.

In: Computational Geometry: Theory and Applications, Vol. 36, No. 2, 02.2007, p. 117-130.

Research output: Contribution to journalArticle

Brass, P, Cenek, E, Duncan, CA, Efrat, A, Erten, C, Ismailescu, DP, Kobourov, SG, Lubiw, A & Mitchell, JSB 2007, 'On simultaneous planar graph embeddings', Computational Geometry: Theory and Applications, vol. 36, no. 2, pp. 117-130. https://doi.org/10.1016/j.comgeo.2006.05.006
Brass, Peter ; Cenek, Eowyn ; Duncan, Cristian A. ; Efrat, Alon ; Erten, Cesim ; Ismailescu, Dan P. ; Kobourov, Stephen G ; Lubiw, Anna ; Mitchell, Joseph S B. / On simultaneous planar graph embeddings. In: Computational Geometry: Theory and Applications. 2007 ; Vol. 36, No. 2. pp. 117-130.
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