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A Constructive Enumeration of Fullerenes
"... In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a topdown approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGIworkstation. The method described can easily be generalised for 3regul ..."
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Cited by 15 (3 self)
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In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a topdown approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGIworkstation. The method described can easily be generalised for 3regular spherical maps with no face having more than 6 edges in its boundary.
Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We disc ..."
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Cited by 9 (0 self)
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A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.
Nonhamiltonian 3connected cubic planar graphs
 SIAM J. Disc. Math
"... We establish that every cyclically 4connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp and we present all nonhamiltonian such graphs of order 42. In addition we list all nonhamiltonian cyclically 5connected cubic planar graphs of orde ..."
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Cited by 6 (3 self)
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We establish that every cyclically 4connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp and we present all nonhamiltonian such graphs of order 42. In addition we list all nonhamiltonian cyclically 5connected cubic planar graphs of order at most 52 and all nonhamiltonian 3connected cubic planar graphs of girth 5 on at most 46 vertices. That all 3connected cubic planar graphs on at most 176 vertices and with face size at most 6 are hamiltonian is also verified. 1 Introduction. In this paper we describe an investigation (making much use of computation) of cyclically kconnected cubic planar graphs (CkCPs) for k = 4, 5 and report the results. We shall also have occasion to consider cubic 3connected planar graphs with no restriction on cyclic connectivity; these we refer to as C3CPs. The investigation
Construction of planar triangulations with minimum degree 5
, 1969
"... In this article we describe a method of constructing all simple triangulations of the sphere with minimum degree 5; equivalently, 3connected planar cubic graphs with girth 5. We also present the results of a computer program based on this algorithm, including counts of convex polytopes of minimum d ..."
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Cited by 3 (1 self)
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In this article we describe a method of constructing all simple triangulations of the sphere with minimum degree 5; equivalently, 3connected planar cubic graphs with girth 5. We also present the results of a computer program based on this algorithm, including counts of convex polytopes of minimum degree 5. Key words: planar triangulation, cubic graph, generation, fullerene
Recent Excluded Minor Theorems
 SURVEYS IN COMBINATORICS, LMS LECTURE NOTE SERIES
"... We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs. ..."
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Cited by 3 (1 self)
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We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs.
Generating rRegular Graphs
, 2002
"... For each nonnegative integer r, we determine a set of graph operations such that all r regular loopless graphs can be generated from the smallest rregular loopless graphs by using these operations. We also discuss possible extensions of this result to rregular graphs of girth at least g, for e ..."
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For each nonnegative integer r, we determine a set of graph operations such that all r regular loopless graphs can be generated from the smallest rregular loopless graphs by using these operations. We also discuss possible extensions of this result to rregular graphs of girth at least g, for each fixed g.
Growing FormFilling Tensegrity Structures using Map LSystems
"... Tensegrities are unique, spacefilling structures consisting of disjoint rigid elements (rods) connected by tensile elements (strings), which hold their shape due to a synergistic balance of opposing forces. Due to their complexity there are few effective analytical methods for discovering new, and ..."
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Tensegrities are unique, spacefilling structures consisting of disjoint rigid elements (rods) connected by tensile elements (strings), which hold their shape due to a synergistic balance of opposing forces. Due to their complexity there are few effective analytical methods for discovering new, and particularly, large irregularly shaped tensegrity structures. Recent efforts using evolutionary search have been moderately successful, but have relied upon a direct encoding of the structure, and therefore face scalability issues [3]. By contrast we employ to a developmental representation grammatically “grow ” tensegrity structures, and as such, issues of scalability, both in terms of representation and of performance, are addressed. Specifically we evolve map Lsystems [4] which produce planar graphs [1, 2] corresponding to the structural