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
A new planarity criterion for 3-connected graphs
β Scribed by Alexander K. Kelmans
- Publisher
- John Wiley and Sons
- Year
- 1981
- Tongue
- English
- Weight
- 474 KB
- Volume
- 5
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
β¦ Synopsis
Abstract
Direct proofs of some planarity criteria are presented.
π SIMILAR VOLUMES
A well-known Tutte's theorem claims that every 3-connected planar graph has a convex embedding into the plane. Tutte's arguments also show that, moreover, for every nonseparating cycle C of a 3-connected graph G, there exists a convex embedding of G such that C is a boundary of the outer face in thi
## Abstract A wellβknown result of Tutte states that a 3βconnected graph __G__ is planar if and only if every edge of __G__ is contained in exactly two induced nonβseparating circuits. Bixby and Cunningham generalized Tutte's result to binary matroids. We generalize both of these results and give n
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)
It is well-known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3-connected planar graph has an edge xy such that deg(x) + deg(y) β€ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we sh
## Abstract We prove that all 3βconnected 4βregular planar graphs can be generated from the Octahedron Graph, using three operations. We generated these graphs up to 15 vertices inclusive. Moreover, by including a fourth operation we obtain an alternative to a procedure by Lehel to generate all con