Numbers of faces in disordered patches

Brinkmann, Gunnar; Graver, Jack E.; Justus, C.

doi:10.1007/s10910-008-9403-6

Cited by 19 publications

(32 citation statements)

References 11 publications

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“…If B k borders an l 1 -loop L 1 and an l 2 -loop L 2 , l 1 , l 2 2, then (a) follows from Lemma 3.7.…”

Section: Fullerenes As Flag Polytopesmentioning

confidence: 98%

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Fullerenes, Polytopes and Toric Topology

Buchstaber¹,

Erokhovets²

2017

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

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The lectures are devoted to a remarkable class of 3-dimensional polytopes, which are mathematical models of the important object of quantum physics, quantum chemistry and nanotechnology -fullerenes. The main goal is to show how results of toric topology help to build combinatorial invariants of fullerenes. Main notions are introduced during the lectures. The lecture notes are addressed to a wide audience.

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“…If B k borders an l 1 -loop L 1 and an l 2 -loop L 2 , l 1 , l 2 2, then (a) follows from Lemma 3.7.…”

Section: Fullerenes As Flag Polytopesmentioning

confidence: 98%

“…Similar argument works for l 1 = 1. Let l 1 , l 2 2. Any edge of γ is an intersection of a facet from L 1 with a facet from L 2 .…”

Section: Fullerenes As Flag Polytopesmentioning

confidence: 99%

Fullerenes, Polytopes and Toric Topology

Buchstaber¹,

Erokhovets²

2017

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

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“…It increases p 6 by one. It was proved in [15] that there is no finite set of growth operations transforming fullerenes into fullerenes sufficient to construct arbitrary fullerene from the dodecahedron. In [12] the example of an infinite set was found.…”

Section: Definitionmentioning

confidence: 99%

“…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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“…These results have been generalized to various problem variants, mainly by varying the conditions on the face lengths and vertex degrees, see e.g. [7,10,11,14]. However, even the question whether it can be decided in polynomial time if at least one hexagonal patch exists with a given boundary code remained open.…”

mentioning

confidence: 99%

Counting Hexagonal Patches and Independent Sets in Circle Graphs

Bonsma

Breuer

2011

Algorithmica

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A hexagonal patch is a plane graph in which inner faces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and threes, how many hexagonal patches exist with this degree sequence along the outer face? This problem is motivated by the enumeration of benzenoid hydrocarbons and fullerenes in computational chemistry. We give the first polynomial time algorithm for this problem. We show that it can be reduced to counting maximum independent sets in circle graphs, and give a simple and fast algorithm for this problem. It is also shown how to subsequently generate hexagonal patches.

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Numbers of faces in disordered patches

Cited by 19 publications

References 11 publications

Fullerenes, Polytopes and Toric Topology

Fullerenes, Polytopes and Toric Topology

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

Counting Hexagonal Patches and Independent Sets in Circle Graphs

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