A generating theorem of punctured surface triangulations with inner degree at least 4
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María-José Chávez
,
Seiya Negami
Abstract
Given any punctured surface F2, we present a method for generating all of F2 triangulations with inner vertices of degree ≥ 4 and boundary vertices of degree ≥ 3. The method is based on a set of expansive operations which includes the well-known vertex splitting and octahedron addition.
By reversing this method we get a procedure to obtain minimal triangulations by a sequence of intermediate triangulations, all of them within the given family.
(Communicated by Peter Horák)
Acknowledgement
The authors would like to thank the anonymous referees for their comments which considerably improved the readability of this paper.
The first and fourth authors are very thankful to Dr. S. Lawrencenko for his helpful comments when preparing this paper and for introducing them to Dr. S. Negami.
References
[1] Barnette, D. W.: Generating the triangulations of the projective plane, J. Combin. Theory Ser. B 33 (1982), 222–230.10.1016/0095-8956(82)90041-7Search in Google Scholar
[2] Barnette, D. W.—Edelson, A. L.: All 2-manifolds have finitely many minimal triangulations, Israel J. Math. 67 (1989), 123–128.10.1007/BF02764905Search in Google Scholar
[3] Batagelj, V.: An improved inductive definition of two restricted classes of triangulations of the plane. Combinatorics and Graph Theory, Banach Center Publ. 25, PWN, Warsaw, 1989, 11–18.10.4064/-25-1-11-18Search in Google Scholar
[4] Boulch, A.—Colin de Verdière, É.—Nakamoto, A.: Irreducible triangulations of surfaces with boundary, Graphs Comb. 29(6) (2013), 1675–1688.10.1007/s00373-012-1244-1Search in Google Scholar
[5] Chávez, M. J.—Lawrencenko, S.—Portillo, J. R.—Villar, M. T.: An algorithm that constructs irreducible triangulations of once-punctured surfaces, http://arXiv:math/arXiv:1410.7738v2.Search in Google Scholar
[6] Chávez, M. J.—Lawrencenko, S.—Quintero, A.—Villar, M. T.: Irreducible triangulations of the Möbius band, Bul. Acad. Sţiinţe Repub. Mold. Mat. 2(75) (2014), 44–50.Search in Google Scholar
[7] Lawrencenko, S.: The irreducible triangulations of the torus, Ukrain. Geom. Sb. 30 (1987), 52–62, (in Russian); J. Soviet Math. 51 (1990), 2537–2543, (Engl. transl.).Search in Google Scholar
[8] Lawrencenko, S.—Negami, S.: Irreducible triangulations of the Klein Bottle, J. Combin. Theory Ser. B 70 (1997), 265–291.10.1006/jctb.1997.9999Search in Google Scholar
[9] Lawrencenko, S.—Sulanke, T.—Villar, M. T.—Zgonnik, L. V.—Chávez, M. J.—Portillo, J. R.: Irreducible triangulations of the once-punctured torus, Sib. Èlektron. Mat. Izv. 15 (2018), 277–304.Search in Google Scholar
[10] Matsumoto, N.—Nakamoto, A.: Generating 4-connected even triangulations on the sphere, Discrete Math. 338 (2015), 64–70.10.1016/j.disc.2014.08.017Search in Google Scholar
[11] Nakamoto, A.—Motoaki, H.: Generating 4-connected triangulations on closed surfaces, Mem. Osaka Kyoiku Univ. Ser. III Nat. Sci. Appl. Sci. 50(2), (2002), 145–153.Search in Google Scholar
[12] Nakamoto, A.—Negami, S.: Generating triangulations on closed surfaces with minimum degree at least 4, Discrete Math. 244 (2002), 345–349.10.1016/S0012-365X(01)00093-0Search in Google Scholar
[13] Negami, S.: Triangulations. In: Handbook of Graph Theory (J. L. Gross, J. Yellen and P. Zhang, eds.), 2nd edition, Chapman and Hall/CRC Press, 2014, 876–901.10.1201/b16132-52Search in Google Scholar
[14] Nishina, M.—Suzuki, Y.: A generating theorem of simple even triangulations with a finitizable set of reductions, Discrete Math. 340 (2017), 2604–2613.10.1016/j.disc.2017.06.018Search in Google Scholar
[15] Steinitz, E.—Rademacher, H.: Vorlesungen über die Theorie der Polyeder, Springer, Berlin, 1934.Search in Google Scholar
[16] Sulanke, T.: Generating irreducible triangulations of surfaces, arXiv:math/0606687v1.Search in Google Scholar
[17] Sulanke, T.: Irreducible triangulations of low genus surfaces, arXiv:math/0606690v1.Search in Google Scholar
[18] Sulanke, T.: Generating maps on surfaces, Discrete Comput. Geom. 57 (2017), 335–356.10.1007/s00454-016-9853-8Search in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- Professor Anatolij Dvurečenskij
- A generating theorem of punctured surface triangulations with inner degree at least 4
- Natural extension of EF-spaces and EZ-spaces to the pointfree context
- On weak isometries in directed groups
- On self-injectivity of the f-ring Frm(𝓟(ℝ), L)
- On the extensions of discrete valuations in number fields
- On certain classes of generalized derivations
- Nilpotent elements in medial semigroups
- Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
- Sharp coefficient bounds for starlike functions associated with the Bell numbers
- Sufficient conditions for Carathéodory functions and applications to univalent functions
- Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks
- Linear directional differential equations in the unit ball: solutions of bounded L-index
- On oscilatory fourth order nonlinear neutral differential equations – IV
- Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms
- Lerch’s theorem on nabla time scales
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- Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula
- Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
- On the amenability of a class of Banach algebras with application to measure algebra
- A bivariate Kumaraswamy-exponential distribution with application
- Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data
- Corrigendum to ”on the congruent number problem over integers of cyclic number fields”, Math. Slovaca 66(3) (2016), 561–564
Articles in the same Issue
- Regular papers
- Professor Anatolij Dvurečenskij
- A generating theorem of punctured surface triangulations with inner degree at least 4
- Natural extension of EF-spaces and EZ-spaces to the pointfree context
- On weak isometries in directed groups
- On self-injectivity of the f-ring Frm(𝓟(ℝ), L)
- On the extensions of discrete valuations in number fields
- On certain classes of generalized derivations
- Nilpotent elements in medial semigroups
- Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
- Sharp coefficient bounds for starlike functions associated with the Bell numbers
- Sufficient conditions for Carathéodory functions and applications to univalent functions
- Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks
- Linear directional differential equations in the unit ball: solutions of bounded L-index
- On oscilatory fourth order nonlinear neutral differential equations – IV
- Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms
- Lerch’s theorem on nabla time scales
-
- Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula
- Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
- On the amenability of a class of Banach algebras with application to measure algebra
- A bivariate Kumaraswamy-exponential distribution with application
- Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data
- Corrigendum to ”on the congruent number problem over integers of cyclic number fields”, Math. Slovaca 66(3) (2016), 561–564
