Irreducible Triangulations of Surfaces with Boundary

Boulch, Alexandre; Verdière, Éric Colin de; Nakamoto, Atsuhiro

doi:10.1007/s00373-012-1244-1

Cited by 13 publications

(29 citation statements)

References 27 publications

(37 reference statements)

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“…An easy consequence is that for fixed b there are only finitely many irreducible braced triangulations. This is analogous to well known results on irreducible triangulations of surfaces by Barnette, Edelson, Boulch, Nakamoto, Colin de Verdiére and others ( [1,3]).…”

Section: Introductionsupporting

confidence: 88%

“…A braced triangulation G = (P, B) is said to be irreducible if there is no edge of P that is contractible in G. This is analogous to the notion of irreducible triangulation that is well studied in the literature on triangulations of surfaces. In that context, it is known that for a given surface there are finitely many isomorphism classes of irreducible triangulationssee [1] and [3]. In this section, we will show that for a given number of braces there are finitely many isomorphism classes of irreducible braced triangulations.…”

Section: Braced Triangulationsmentioning

confidence: 85%

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Braced triangulations and rigidity

Cruickshank¹,

Kastis²,

Kitson³

et al. 2021

Preprint

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We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer b there is such an inductive construction of triangulations with b braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with b braces that is linear in b. In the case that b = 1 or 2 we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are (generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in R 4 and a class of mixed norms on R 3 . 2020 Mathematics Subject Classification. 52C25, 05C10, 46B20. E.K. and D.K.

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Section: Introductionsupporting

confidence: 88%

Section: Braced Triangulationsmentioning

confidence: 85%

Braced triangulations and rigidity

Cruickshank¹,

Kastis²,

Kitson³

et al. 2021

Preprint

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show abstract

“…(This result has been extended to surfaces with boundaries by Boulch and al. [BCdVN13]. The best upper bound known to date is due to Joret and Wood [JW10] who proved that this number is at most max{13g − 4, 4}.)…”

Section: State Of the Art 31 Splitting Cycles On Triangulationsmentioning

confidence: 99%

Some Triangulated Surfaces without Balanced Splitting

Despré

Lazarus

2016

Graphs and Combinatorics

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Let G be the graph of a triangulated surface Σ of genus g ≥ 2. A cycle of G is splitting if it cuts Σ into two components, neither of which is homeomorphic to a disk. A splitting cycle has type k if the corresponding components have genera k and g − k. It was conjectured that G contains a splitting cycle (Barnette '1982). We confirm this conjecture for an infinite family of triangulations by complete graphs but give counter-examples to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should contain splitting cycles of every possible type.

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“…A triangulation G is irreducible if G has no contractible edge. It is known that every surface admits finitely many irreducible triangulations, up to homeomorphism , and the complete lists of irreducible triangulations are known for

S_{0}

S_{1}

S_{2}

N_{1}

N_{2}

N_{3}

, and

N_{4}

. (See for more details about irreducible triangulations.…”

Section: Applicationmentioning

confidence: 99%

Even Embeddings of the Complete Graphs and Their Cycle Parities

Noguchi

2016

Journal of Graph Theory

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The complete graph Kn on n vertices can be quadrangularly embedded on an orientable (resp. nonorientable) closed surface F2 with Euler characteristic εfalse(F2false)=nfalse(5−nfalse)/4 if and only if n≡0,5(mod8) (resp. n≡0,1(mod4) and n≠5). In this article, we shall show that if Kn quadrangulates a closed surface F2, then Kn has a quadrangular embedding on F2 so that the length of each closed walk in the embedding has the parity specified by any given homomorphism ρ:π1false(F2false)→double-struckZ2, called the cycle parity.

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Irreducible Triangulations of Surfaces with Boundary

Cited by 13 publications

References 27 publications

Braced triangulations and rigidity

Braced triangulations and rigidity

Some Triangulated Surfaces without Balanced Splitting

Even Embeddings of the Complete Graphs and Their Cycle Parities

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