On generating planar graphs

Barnette, David

doi:10.1016/s0012-365x(74)80001-4

Cited by 7 publications

(11 citation statements)

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“…For every graph G in G, there is a sequence G 0 , G 1 , ..., G t of members of G such that G 0 is the tetrahedron, G t is G, and each G i+1 is obtained from G i by adding a handle. In [1,2,4,12], analogous results are obtained for 3-regular simple planar graphs with other connectivities. For 4-regular simple planar graphs, the situation is similar and the readers are referred to [3,9,10].…”

Section: Introductionmentioning

confidence: 62%

Generating 5‐regular planar graphs

Ding

Kanno

2009

Journal of Graph Theory

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Section: Introductionmentioning

confidence: 62%

Generating 5‐regular planar graphs

Ding

Kanno

2009

Journal of Graph Theory

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“…Since v 4 is in the exterior of Δ 1 and v 3 is in the exterior of Δ 2 , the convex hull ch(v 1 4 ) in counterclockwise order. By Lemma 4, T 1 (resp., T 2 ) has a flippable edge e 1 (resp., e 2 ) in a triangle opposite to s. If flipping edge e 1 in T 1 or edge e 2 in T 2 increases the degree of s to 5, then perform the edge flip.…”

Section: Lemma 3 [18] Let G = (S E) Be a Maximal Biplane Graph Inmentioning

confidence: 99%

Geometric Biplane Graphs II: Graph Augmentation

García

Hurtado²,

Korman

et al. 2015

Graphs and Combinatorics

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We study biplane graphs drawn on a finite point set $S$ in the plane in general position. This is the family of geometric graphs whose vertex set is $S$ and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges

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“…We have now generated all C5CPs on up Downloaded 05/11/18 to 34.218.44.141. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php to 52 vertices using the generation procedure of Barnette [1] and Butler [6]. The graphs were then checked for hamiltonian cycles.…”

Section: Theorem 2 If G Is a Nonhamiltonian C4cp Of Smallest Order Amentioning

confidence: 99%

Nonhamiltonian 3-Connected Cubic Planar Graphs

Aldred¹,

Bau²,

Holton³

et al. 2000

SIAM J. Discrete Math.

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Abstract. We establish that every cyclically 4-connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp, and we present all nonhamiltonian examples of order 42. In addition we list all nonhamiltonian cyclically 5-connected cubic planar graphs of order at most 52 and all nonhamiltonian 3-connected cubic planar graphs of girth 5 on at most 46 vertices. The fact that all 3-connected cubic planar graphs on at most 176 vertices and with face size at most 6 are hamiltonian is also verified.Key words. nonhamiltonian, cubic, planar AMS subject classification. 05C38 PII. S0895480198348665 1. Introduction. In this paper we describe an investigation (making much use of computation) of cyclically k-connected cubic planar graphs (CkCP s) for k = 4, 5 and report the results. We shall also have occasion to consider cubic 3-connected planar graphs with no restriction on cyclic connectivity; these we refer to as C3CP s. The investigation extends the work of Holton and McKay in [10], in which the smallest order of a nonhamiltonian C3CP was shown to be 38. We provide answers to two questions raised in that paper, namely, the following:(a) What is the smallest order of a nonhamiltonian C4CP? (b) Is there more than one nonhamiltonian C5CP on 44 vertices? The answer to the former question is determined to be 42, and all nonhamiltonian C4CPs of that order are presented. The latter question is answered in the negative, and a complete list of all nonhamiltonian C5CPs on at most 52 vertices is presented.Before proceeding, we include some definitions and results from the existing literature as we shall make use of them in the rest of the paper. By a k-gon we mean a face of a plane graph bounded by k edges. Note that a k-cycle is not necessarily a k-gon. By a k-cut we mean a set of k edges whose removal leaves the graph disconnected and of which no proper subset has that property. The two components (and clearly there are only two) formed by the removal of a k-cut are called k-pieces. A k-cut is nontrivial if each of its k-pieces contains a cycle and is essential if it is nontrivial and each of its k-pieces contains more than k vertices. A cubic graph is cyclically k-connected if it has no nontrivial t-cuts for 0 ≤ t ≤ k − 1, and it has cyclic connectivity k if, in addition, it has at least one nontrivial k-cut. We denote by λ (G) the value of k such that the C3CP G has cyclic connectivity k.If G is a hamiltonian C3CP, then an a-edge is an edge which is present in every hamiltonian cycle in G, while a b-edge is absent from every hamiltonian cycle in G.To focus our search for nonhamiltonian C4CPs of smallest order we shall make use of the following theorem which formed the main result in [10].

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On generating planar graphs

Cited by 7 publications

References 0 publications

Generating 5‐regular planar graphs

Generating 5‐regular planar graphs

Geometric Biplane Graphs II: Graph Augmentation

Nonhamiltonian 3-Connected Cubic Planar Graphs

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