Irreducible triangulations are small

Joret, Gwenaël; Wood, David R.

doi:10.1016/j.jctb.2010.01.004

Cited by 14 publications

(24 citation statements)

References 23 publications

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“…Our method does not improve over the current best bound of max{13g −4, 4} by Joret and Wood [15]. However, it is substantially different and simpler than the other known proofs of this Starting with a set of triangles glued together, all meeting at a vertex (bottom part), attach a set of g/2 pairs of interlaced rectangular strips (top left) and a set of b − 1 non-interlaced rectangular strips (top right), and triangulate every strip by adding an arbitrary diagonal (not shown in the picture).…”

Section: Our Resultsmentioning

confidence: 70%

“…Finally, we refine the above technique in the case of surfaces without boundary, and obtain a bound that is better than that of Theorem 1, but no better than the current best result by Joret and Wood [15].…”

Section: Our Resultsmentioning

confidence: 84%

“…Nakamoto and Ota [26] were the first to show that the number of vertices in an irreducible triangulation is at most linear in the genus of the surface. The best upper bound known to date is due to Joret and Wood [15], who proved that this number is at most max{13g − 4, 4}. (Here and in the sequel, g is the Euler genus, which equals twice the usual genus for orientable surfaces and equals the usual genus for non-orientable surfaces.)…”

Section: Previous Work For Surfaces Without Boundarymentioning

confidence: 99%

“…Negami [27,28] used the fact that there are finitely many irreducible triangulations to prove that two triangulations of a surface with the same number of vertices are equivalent under diagonal flips, provided the number of vertices is greater than an integer depending only on the surface. For further applications, see the recent paper by Joret and Wood [15] and references therein.…”

Section: Applications Of Irreducible Triangulationsmentioning

confidence: 99%

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Irreducible Triangulations of Surfaces with Boundary

Boulch

Verdière

Nakamoto

2012

Graphs and Combinatorics

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A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g ≥ 0 with b ≥ 0 boundary components is O (g + b). So far, the result was known only for surfaces without boundary (b = 0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary.

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Section: Our Resultsmentioning

confidence: 70%

Section: Our Resultsmentioning

confidence: 84%

Section: Previous Work For Surfaces Without Boundarymentioning

confidence: 99%

Section: Applications Of Irreducible Triangulationsmentioning

confidence: 99%

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Irreducible Triangulations of Surfaces with Boundary

Boulch

Verdière

Nakamoto

2012

Graphs and Combinatorics

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“…To 286 the best of our knowledge, the best known upper bound on the size of irreducible triangulations was given by Jaret and Wood [29]:…”

Section: Of T Is Non-contractible Then T Is Called Irreducible; Elsementioning

confidence: 99%

A fast algorithm for computing irreducible triangulations of closed surfaces in Ed

Ramaswami

Siqueira

2018

Computational Geometry

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We give a fast algorithm for computing an irreducible triangulation T ′ of an oriented, connected, boundaryless, and compact surface S in E d from any given triangulation T of S. If the genus g of S is positive, then our algorithm takes O(g 2 +gn) time to obtain T ′ , where n is the number of triangles of T . Otherwise, T ′ is obtained in linear time in n. While the latter upper bound is optimal, the former upper bound improves upon the currently best known upper bound by a lg n/g factor. In both cases, the memory space required by our algorithm is in Θ(n).

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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 are small

Cited by 14 publications

References 23 publications

Irreducible Triangulations of Surfaces with Boundary

Irreducible Triangulations of Surfaces with Boundary

A fast algorithm for computing irreducible triangulations of closed surfaces in Ed

Even Embeddings of the Complete Graphs and Their Cycle Parities

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