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A Characterization of Almost-Planar Graphs

  • Bradley S. Gubser (a1)
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.

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[2]Bondy, J. A. and Murty, U. S. R. (1976) Graph Theory with Applications. North-Holland.
[3]Hall, D. W. (1943) A note on primitive skew curves. Bull. Amer. Math. Soc. 49 935937.
[4]Kuratowski, K. (1930) Sur le problème des courbes gauches en topologie. Fund. Math. 15 271283.
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[7]Wagner, K. (1937) Über eine eigenshaft der ebenen komplexe. Math. Ann. 114 570590.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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