Geometric Biplane Graphs I: Maximal Graphs
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.
KeywordsGeometric graphs Biplane graphs Maximal biplane graphs \(k\)-Connected graphs Graph augmentation
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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