Abstract
In this paper we prove, among a few other results, that if G is a connected quadrangulation of the sphere with minimum degree 3 and with no separating quadrilateral then G is 3-connected.
Dedicated to the 70th anniversary of Professor Tudor Zamfirescu
Work supported by a Competitive Program for Rate Researchers (CPRR), NRF South Africa.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
V. Batagelj, An inductive definition of the class of 3-connected quadrangulations of the plane. Discrete Math. 78, 45–53 (1989)
S. Bau, Contraction of a closed neighbourhood in bipartite plane graphs. Mong. Math. J. 18, 7–12 (2014)
S. Bau, N. Matsumoto, A. Nakamoto, L-J. Zheng, Minor relations for quadrangulations of the sphere, to appear in Graphs and Combinatorics
G. Brinkmann, S. Greenberg, C. Greenhill, B.D. McKay, R. Thomas, P. Wollan, Generation of simple quadrangulations of the sphere. Discrete Math. 305, 33–54 (2005)
H.J. Broersma, A.J.W. Duijvestijn, F. Göbel, Generating all 3-connected 4-regular planar graphs from the octahedron graph. J. Graph Theory 17, 613–620 (1993)
R. Diestel, Graph Theory (Springer, New York, 1997)
M. Kriesell, A constructive characterization of 3-connected triangle-free graphs. J. Combin. Theory B 97, 358–370 (2007)
W. Mader, On \(k\)-critically \(n\)-connected graphs, in Progress in Graph Theory ed. by J.A. Bondy, U.S.R. Murty (Academic Press, Orlando, 1984), pp. 389–398
N. Martinov, Uncontractible 4-connected graphs. J. Graph Theory 6, 343–344 (1982)
N. Martinov, A recursive characterization of the 4-connected graphs. Discrete Math. 84, 105–108 (1990)
A. Nakamoto, Generating quadrangulations of surfaces with minimum degree at least 3. J. Graph Theory 30, 223–234 (1999)
A. Saito, Splitting and contractible edges in 4-connected graphs. J. Combin. Theory B 88, 227–235 (2003)
E. Steinitz, H. Rademacher, Vorlesungen über die Theorie der Polyeder (Springer, Berlin, 1934)
W.T. Tutte, A theory of 3-connected graphs. Indag. Math. 23, 441–455 (1961)
Acknowledgments
The author is indebted to an anonymous referee whose comments improved the paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Bau, S. (2016). Reductions of 3-Connected Quadrangulations of the Sphere. In: Adiprasito, K., Bárány, I., Vilcu, C. (eds) Convexity and Discrete Geometry Including Graph Theory. Springer Proceedings in Mathematics & Statistics, vol 148. Springer, Cham. https://doi.org/10.1007/978-3-319-28186-5_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-28186-5_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28184-1
Online ISBN: 978-3-319-28186-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)