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Geometric Biplane Graphs I: Maximal Graphs

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Abstract

We study biplane graphs drawn on a finite planar point set \(S\) in general position. This is the family of geometric graphs whose vertex set is \(S\) and can be decomposed into two plane graphs. We show that two maximal biplane graphs—in the sense that no edge can be added while staying biplane—may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over \(n\)-element point sets.

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Notes

  1. We note that Eppstein’s algorithm does not explicitly construct the intersection graph \(G_X\), which may have \(\Omega (m^2)\) edges.

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Acknowledgments

A. G., F. H., M. K., R.I. S. and J. T. were partially supported by ESF EUROCORES Programme EuroGIGA, CRP ComPoSe: Grant EUI-EURC-2011-4306, and by Project MINECO MTM2012-30951/FEDER. F. H., and R.I. S. were also supported by Project Gen. Cat. DGR 2009SGR1040. A. G. and J. T. were also supported by Project E58-DGA. M. K. was supported by the Secretary for Universities and Research of the Ministry of Economy and Knowledge of the Government of Catalonia and the European Union. I. M. was supported by FEDER funds through COMPETE–Operational Programme Factors of Competitiveness, CIDMA and FCT within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690. M. S. was supported by the Project NEXLIZ - CZ.1.07/2.3.00/30.0038, which is co-financed by the European Social Fund and the state budget of the Czech Republic, and by ESF EuroGIGA Project ComPoSe as F.R.S.-FNRS—EUROGIGA NR 13604. R. S. was funded by Portuguese Funds through CIDMA (Center for Research and Development in Mathematics and Applications) and FCT (Fundação para a Ciência e a Tecnologia), within Project PEst-OE/MAT/UI4106/2014, and by FCT Grant SFRH/BPD/88455/2012.

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Authors and Affiliations

  1. Departamento de Métodos Estadísticos, IUMA, Universidad de Zaragoza, Zaragoza, Spain

    Alfredo García & Javier Tejel

  2. Departament de Matemàtica Aplicada II, UPC, Barcelona, Spain

    Ferran Hurtado & Rodrigo I. Silveira

  3. Erato Kawarabayashi Large Graph Project, JST, National Institute of Informatics (NII), Tokyo, Japan

    Matias Korman

  4. Departamento de Matemática and CIDMA, Universidade de Aveiro, Aveiro, Portugal

    Inês Matos

  5. Department of Mathematics and European Centre of Excellence NTIS (New Technologies for the Information Society), University of West Bohemia, Pilsen, Czech Republic

    Maria Saumell

  6. Department of Mathematics, California State University Northridge, Los Angeles, USA

    Csaba D. Tóth

Authors
  1. Alfredo García
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  2. Ferran Hurtado
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  3. Matias Korman
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  4. Inês Matos
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  5. Maria Saumell
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  6. Rodrigo I. Silveira
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  7. Javier Tejel
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  8. Csaba D. Tóth
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Corresponding author

Correspondence to Alfredo García.

Additional information

A preliminary version of this paper has been presented at the Mexican Conference on Discrete Mathematics and Computational Geometry, Oaxaca, México, November 2013 [14].

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García, A., Hurtado, F., Korman, M. et al. Geometric Biplane Graphs I: Maximal Graphs. Graphs and Combinatorics 31, 407–425 (2015). https://doi.org/10.1007/s00373-015-1546-1

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  • DOI: https://doi.org/10.1007/s00373-015-1546-1

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