Geometric Biplane Graphs II: Graph Augmentation

We study biplane graphs drawn on a finite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there ar...

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Published in:	Graphs and combinatorics Vol. 31; no. 2; pp. 427 - 452
Main Authors:	García, Alfredo, Hurtado, Ferran, Korman, Matias, Matos, Inês, Saumell, Maria, Silveira, Rodrigo I., Tejel, Javier, Tóth, Csaba D.
Format:	Journal Article Publication
Language:	English
Published:	Tokyo Springer Japan 01.03.2015 Springer Nature B.V
Subjects:	30 Functions of a complex variable 30C Geometric function theory Augmentation Biplane graphs Biplanes Classificació AMS Combinatorial analysis Combinatorics COMMON ANCESTORS CONNECTIVITY Engineering Design Geometria Geometria computacional Geometric graphs Geometric group theory Geometry Graph augmentation Graph theory Graphs k-connected graphs Matemàtiques i estadística Mathematical analysis Mathematics Mathematics and Statistics Original Paper Planes Texts Wheels Àrees temàtiques de la UPC Biplane graphs connected graphs Geometric graphs Graph augmentation
ISSN:	0911-0119, 1435-5914
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Abstract	We study biplane graphs drawn on a finite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.
AbstractList	(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We study biplane graphs drawn on a finite point set ... in the plane in general position. This is the family of geometric graphs whose vertex set is ... and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges. We study biplane graphs drawn on a finite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Issue Title: Special Issue: Combinatorics and Geometry We study biplane graphs drawn on a finite point set ... in the plane in general position. This is the family of geometric graphs whose vertex set is ... and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges. We study biplane graphs drawn on a finite point set in the plane in general position. This is the family of geometric graphs whose vertex set is and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges. Peer Reviewed
Author	Korman, Matias Tejel, Javier Silveira, Rodrigo I. Matos, Inês Tóth, Csaba D. Saumell, Maria García, Alfredo Hurtado, Ferran
Author_xml	– sequence: 1 givenname: Alfredo surname: García fullname: García, Alfredo organization: Departamento de Métodos Estadísticos, IUMA, Universidad de Zaragoza – sequence: 2 givenname: Ferran surname: Hurtado fullname: Hurtado, Ferran organization: Departament de Matemàtica Aplicada II, UPC – sequence: 3 givenname: Matias surname: Korman fullname: Korman, Matias organization: Erato Kawarabayashi Large Graph Project, JST, National Institute of Informatics (NII) – sequence: 4 givenname: Inês surname: Matos fullname: Matos, Inês organization: Departamento de Matemática and CIDMA, Universidade de Aveiro – sequence: 5 givenname: Maria surname: Saumell fullname: Saumell, Maria organization: Department of Mathematics and European Centre of Excellence NTIS (New Technologies for the Information Society), University of West Bohemia – sequence: 6 givenname: Rodrigo I. surname: Silveira fullname: Silveira, Rodrigo I. organization: Departament de Matemàtica Aplicada II, UPC – sequence: 7 givenname: Javier surname: Tejel fullname: Tejel, Javier email: jtejel@unizar.es organization: Departamento de Métodos Estadísticos, IUMA, Universidad de Zaragoza – sequence: 8 givenname: Csaba D. surname: Tóth fullname: Tóth, Csaba D. organization: Department of Mathematics, California State University Northridge
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CitedBy_id	crossref_primary_10_1016_j_amc_2020_125513 crossref_primary_10_1016_j_comgeo_2021_101819
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Contributor	Universitat Politècnica de Catalunya. DCCG - Grup de recerca en geometria computacional, combinatoria i discreta Universitat Politècnica de Catalunya. Departament de Matemàtiques
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Theory Appl. doi: 10.1016/j.comgeo.2007.09.001
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SubjectTerms	30 Functions of a complex variable 30C Geometric function theory Augmentation Biplane graphs Biplanes Classificació AMS Combinatorial analysis Combinatorics COMMON ANCESTORS CONNECTIVITY Engineering Design Geometria Geometria computacional Geometric graphs Geometric group theory Geometry Graph augmentation Graph theory Graphs k-connected graphs Matemàtiques i estadística Mathematical analysis Mathematics Mathematics and Statistics Original Paper Planes Texts Wheels Àrees temàtiques de la UPC
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Title	Geometric Biplane Graphs II: Graph Augmentation
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