## Abstract Direct proofs of some planarity criteria are presented.
A strengthening of the Kuratowski planarity criterion for 3-connected graphs
β Scribed by A.K. Kelmans
- Publisher
- Elsevier Science
- Year
- 1984
- Tongue
- English
- Weight
- 617 KB
- Volume
- 51
- Category
- Article
- ISSN
- 0012-365X
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β¦ Synopsis
Re&ved 4 Fkbruary
π SIMILAR VOLUMES
We present a new short combinatorial proof of the sufficiency part of the well-known Kuratowski's graph planarity criterion. The main steps are to prove that for a minor minimal non-planar graph G and any edge xy: (1) G-x-y does not contain ΞΈ-subgraph; (2) G-x-y is homeomorphic to the circle; (3)
In a recent paper, Carsten Thomassen [Carsten Thomassen, Planarity and duality of finite and infinite graphs. J. Combinatorial Theory Ser. B 29 (1980) 244-2711 has shown that a number of criteria for the planarity of a graph can be reduced to that of Kuratowski. Here we present another criterion whi
## Abstract Let __C__ be a longest cycle in the 3βconnected graph __G__ and let __H__ be a component of __G__βββ__V__(__C__) such that |__V__(__H__)|ββ₯β3. We supply estimates of the form |__C__|ββ₯β2__d__(__u__)β+β2__d__(__v__)βββΞ±(4ββ€βΞ±ββ€β8), where __u__,__v__ are suitably chosen nonβadjacent verti