A Catalogue of Growth Transformations of Fullerene Polyhedra.

Brinkmann, Gunnar; Fowler, Patrick W.

doi:10.1002/chin.200405236

Cited by 2 publications

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“…No explicit complexity bounds are given in [3] and [7], but we observe that the worst case complexity is superexponential, a rough bound is O n n 2 . Other algorithms use variants of this branching approach that apply to sequences S of a special form [6,4], or simply generate all possible patches and categorize them according to boundary codes [5].…”

Section: Theorem 2 the Number Of Hexagonal Patches That Satisfy A Boumentioning

confidence: 99%

“…For detailed definitions see also Section 2. In graph theoretical terms, fullerenes can be modelled by 3-regular plane graphs with only 5-faces and 6-faces (fullerene graphs). In the study of how fullerenes are generated and can be enumerated, the following concept is essential [9,5,4]: a fullerene patch can be obtained from a fullerene graph by taking a cycle in the plane graph and removing every vertex and edge outside of the cycle. This motivates the following definition: a fullerene patch (or simply patch) is a 2-connected plane graph in which every inner face has length 5 or 6, every non-boundary vertex has degree 3, and boundary vertices have degree 2 or 3.…”

Section: Introductionmentioning

confidence: 99%

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Finding Fullerene Patches in Polynomial Time

Bonsma

Breuer

2009

Algorithms and Computation

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Abstract. We consider the following question, motivated by the enumeration of fullerenes. A fullerene patch is a 2-connected plane graph G in which inner faces have length 5 or 6, non-boundary vertices have degree 3, and boundary vertices have degree 2 or 3. The degree sequence along the boundary is called the boundary code of G. We show that the question whether a given sequence S is a boundary code of some fullerene patch can be answered in polynomial time when such patches have at most five 5-faces. We conjecture that our algorithm gives the correct answer for any number of 5-faces, and sketch how to extend the algorithm to the problem of counting the number of different patches with a given boundary code.

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Section: Theorem 2 the Number Of Hexagonal Patches That Satisfy A Boumentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Finding Fullerene Patches in Polynomial Time

Bonsma

Breuer

2009

Algorithms and Computation

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“…Thus, the important problem is to define and study different structures and operations on the set of all combinatorial types of fullerenes. The well-known problem [9,11,12,13,14,15,16] is to find a simple set of operations sufficient to construct arbitrary fullerene from the dodecahedron C 20 .…”

Section: Introductionmentioning

confidence: 99%

Finite sets of operations sufficient to construct any fullerene from C 20

Buchstaber

Erokhovets

2016

Struct Chem

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We study the well-known problem of combinatorial classification of fullerenes. By a (mathematical) fullerene we mean a convex simple three dimensional polytope with all facets pentagons and hexagons. We analyse approaches of construction of arbitrary fullerene from the dodecahedron (a fullerene C 20 ). A growth operation is a combinatorial operation that substitutes the patch with more facets and the same boundary for the patch on the surface of a simple polytope to produce a new simple polytope. It is known that an infinite set of different growth operations transforming fullerenes into fullerenes is needed to construct any fullerene from the dodecahedron. We prove that if we allow a polytope to contain one exceptional facet, which is a quadrangle or a heptagon, then a finite set of growth operation is sufficient. We analyze pairs of objects: a finite set of operations, and a family of acceptable polytopes containing fullerenes such that any polytope of the family can be obtained from the dodecahedron by a sequence of operations from the corresponding set. We describe explicitly three such pairs. First two pairs contain seven operations, and the last -eleven operations. Each of these operations corresponds to a finite set of growth operations and is a composition of edge-and two edges-truncations.

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A Catalogue of Growth Transformations of Fullerene Polyhedra.

Cited by 2 publications

References 1 publication

Finding Fullerene Patches in Polynomial Time

Finding Fullerene Patches in Polynomial Time

Finite sets of operations sufficient to construct any fullerene from C 20

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