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A Characterization of Almost-Planar Graphs

Published online by Cambridge University Press:  12 September 2008

Bradley S. Gubser
Bradley S. Gubser
Affiliation:
Department of Mathematics, Hiram College, Hiram, OH 44234, USA Email: gubserbs@hirama.hiram.edu
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Abstract

Kuratowski's Theorem, perhaps the most famous result in graph theory, states that K5 and K3,3 are the only non-planar graphs for which both G\e, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the almost-planar graphs, those non-planar graphs for which G\e or G/e is planar for all edges e of G. This paper gives two characterizations of the almost-planar graphs: an explicit description of the structure of almost-planar graphs; and an excluded minor criterion. We also give a best possible bound on the number of edges of an almost-planar graph.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 3 , September 1996 , pp. 227 - 245
Copyright
Copyright © Cambridge University Press 1996

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References

[1]Bergmann, H. (1982) Ein Planaritätskriterium für endliche Graphen. Elem. Math. 37 49–51.Google Scholar
[2]Bondy, J. A. and Murty, U. S. R. (1976) Graph Theory with Applications. North-Holland.CrossRefGoogle Scholar
[3]Hall, D. W. (1943) A note on primitive skew curves. Bull. Amer. Math. Soc. 49 935–937.CrossRefGoogle Scholar
[4]Kuratowski, K. (1930) Sur le problème des courbes gauches en topologie. Fund. Math. 15 271–283.CrossRefGoogle Scholar
[5]Negami, S. (1982) A characterization of 3-connected graphs containing a given graph. J. Combin. Theory 32 69–74.CrossRefGoogle Scholar
[6]Seymour, P. D. (1980) Decomposition of regular matroids. J. Combin. Theory Ser. 528 305–359.CrossRefGoogle Scholar
[7]Wagner, K. (1937) Über eine eigenshaft der ebenen komplexe. Math. Ann. 114 570–590.CrossRefGoogle Scholar