Discrete & Computational Geometry

, Volume 53, Issue 3, pp 587–620 | Cite as

Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

  • Éric Colin de Verdière
  • Alfredo Hubard
  • Arnaud de Mesmay
Article

Abstract

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart). Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki (Graph structure theory. Contemporary Mathematics, vol. 147, 1993), a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov’s systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions. Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length \(O(g^{3/2}n^{1/2})\) for any triangulated combinatorial surface of genus \(g\) with \(n\) triangles, and describe an \(O(gn)\)-time algorithm to compute such a decomposition. Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.

Keywords

Systole Edge-width Surface Closed curve  Cut graph  Pants decomposition 

Mathematics Subject Classification

05C10 68U05 53C23 57M15 68R10 

References

  1. 1.
    Balacheff, F., Parlier, H., Sabourau, S.: Short loop decompositions of surfaces and the geometry of Jacobians. Geom. Funct. Anal. 22(1), 37–73 (2012)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Boissonnat, J.-D., Dyer, R., Ghosh, A.: Delaunay triangulations of manifolds. arXiv:1311.0117 (2013)
  3. 3.
    Brooks, R., Makover, E.: Random construction of Riemann surfaces. J. Differ. Geom. 68(1), 121–157 (2004)MATHMathSciNetGoogle Scholar
  4. 4.
    Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Progress in Mathematics, vol. 106. Birkhäuser, Basel (1992)Google Scholar
  5. 5.
    Buser, P., Sarnak, P.: On the period matrix of a Riemann surface of large genus (with an appendix by J.H. Conway and N.J.A. Sloane). Invent. Math. 117, 27–56 (1994)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Cabello, S., Mohar, B.: Finding shortest non-separating and non-contractible cycles for topologically embedded graphs. Discrete Comput. Geom. 37(2), 213–235 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Cabello, S., Colin de Verdière, É., Lazarus, F.: Algorithms for the edge-width of an embedded graph. Comput. Geom. 45, 215–224 (2012)Google Scholar
  8. 8.
    Cabello, S., Chambers, E.W., Erickson, J.: Multiple-source shortest paths in embedded graphs. SIAM J. Comput. 42(4), 1542–1571 (2013)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chambers, E.W., Colin de Verdière, É., Erickson, J., Lazarus, F., Whittlesey, Kim: Splitting (complicated) surfaces is hard. Comput. Geom. 41(1–2), 94–110 (2008)Google Scholar
  10. 10.
    Chambers, E., Erickson, J., Nayyeri, A.: Minimum cuts and shortest homologous cycles. In: Proceedings of the 25th Annual Symposium on Computational Geometry (SOCG), pp. 377–385. ACM, New York (2009)Google Scholar
  11. 11.
    Chambers, E.W., Erickson, J., Nayyeri, A.: Homology flows, cohomology cuts. SIAM J. Comput. 41(6), 1605–1634 (2012)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Colin de Verdière, É.: Topological algorithms for graphs on surfaces. PhD thesis, École normale supérieure, 2012. Habilitation thesis, available at http://www.di.ens.fr/colin/
  13. 13.
    Colin de Verdière, É., Erickson, J.: Tightening nonsimple paths and cycles on surfaces. SIAM J. Comput. 39(8), 3784–3813 (2010)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Colin de Verdière, É., Lazarus, F.: Optimal pants decompositions and shortest homotopic cycles on an orientable surface. J. ACM 54(4): Article 18 (2007)Google Scholar
  15. 15.
    do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)CrossRefMATHGoogle Scholar
  16. 16.
    Dyer, R., Zhang, H., Möller, T.: Surface sampling and the intrinsic Voronoi diagram. Comput. Graph. Forum 27(5), 1393–1402 (2008)CrossRefGoogle Scholar
  17. 17.
    Eppstein, D.: Squarepants in a tree: sum of subtree clustering and hyperbolic pants decomposition. ACM Trans. Algorithms 5(3), 1–24 (2009)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Erickson, J.: Combinatorial optimization of cycles and bases. In: Zomorodian, A. (ed.) Computational Topology. In: Proceedings of Symposia in Applied Mathematics. American Mathematical Society, Providence, RI (2012)Google Scholar
  19. 19.
    Erickson, J., Har-Peled, S.: Optimally cutting a surface into a disk. Discrete Comput. Geom. 31(1), 37–59 (2004)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Erickson, J., Nayyeri, A.: Computing replacement paths in surface-embedded graphs. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1347–1354 (2011)Google Scholar
  21. 21.
    Erickson, J., Whittlesey, K.: Greedy optimal homotopy and homology generators. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1038–1046 (2005)Google Scholar
  22. 22.
    Erickson, J., Worah, P.: Computing the shortest essential cycle. Discrete Comput. Geom. 44(4), 912–930 (2010)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Erickson, J., Fox, K., Nayyeri, A.: Global minimum cuts in surface embedded graphs. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1309–1318 (2012)Google Scholar
  24. 24.
    Gamburd, A., Makover, E.: On the genus of a random riemann surface. Complex Manifolds and Hyperbolic Geometry. Contemporary Mathematics, vol. 311, pp. 133–140. American Mathematical Society, Providence, RI (2002)CrossRefGoogle Scholar
  25. 25.
    Geelen, J., Huynh, T., Richter, R.B.: Explicit bounds for graph minors. arxiv:1305.1451 (2013)
  26. 26.
    Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom. 18, 1–147 (1983)MATHMathSciNetGoogle Scholar
  27. 27.
    Gromov, M.: Systoles and intersystolic inequalities. In: Actes de la table ronde de géométrie différentielle, pp. 291–362 (1992)Google Scholar
  28. 28.
    Gromov, M.: Sign and geometric meaning of curvature. Rend. Semin. Mat. Fis. Milano 61(1991), 9–123 (1994)MathSciNetGoogle Scholar
  29. 29.
    Guskov, I., Wood, Z.J.: Topological noise removal. In: Proceedings of Graphics Interface, pp. 19–26 (2001)Google Scholar
  30. 30.
    Guth, L., Parlier, H., Young, R.: Pants decompositions of random surfaces. Geom. Funct. Anal. 21, 1069–1090 (2011)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Han, Q., Hong, J.-X.: Isometric Embedding of Riemannian Manifolds in Euclidean Spaces. Mathematical Surveys and Monographs, vol. 130. American Mathematical Society, Providence, RI (2006)CrossRefGoogle Scholar
  32. 32.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002). Available at http://www.math.cornell.edu/hatcher/
  33. 33.
    Hutchinson, J.P.: On short noncontractible cycles in embedded graphs. SIAM J. Discrete Math. 1(2), 185–192 (1988)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Katz, M.: Systolic Geometry and Topology. Mathematical Surveys and Monographs, vol. 137. American Mathematical Society, Providence, RI (2007). (With an appendix by J. Solomon.)CrossRefGoogle Scholar
  35. 35.
    Katz, M.G., Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds. Ergodic Theory Dyn. Syst. 25(4), 1209–1220 (2005)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Kervaire, M.A.: A manifold which does not admit any differentiable structure. Comment. Math. Helv. 34, 257–270 (1960)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Klingenberg, W.: Riemannian Geometry. Philosophie und Wissenschaft. de Gruyter, Berlin (1995)CrossRefGoogle Scholar
  38. 38.
    Kowalick, R.: Discrete systolic inequalities. PhD Thesis, Ohio State University (2013). http://rave.ohiolink.edu/etdc/view?acc_num=osu1384873457
  39. 39.
    Kutz, M.: Computing shortest non-trivial cycles on orientable surfaces of bounded genus in almost linear time. In: Proceedings of the 22nd Annual Symposium on Computational Geometry (SOCG), pp. 430–438. American Mathematical Society, Providence, RI (2006)Google Scholar
  40. 40.
    Lazarus, F., Pocchiola, M., Vegter, G., Verroust, A.: Computing a canonical polygonal schema of an orientable triangulated surface. In: Proceedings of the 17th Annual Symposium on Computational Geometry (SOCG), pp. 80–89. American Mathematical Society, Providence, RI (2001)Google Scholar
  41. 41.
    Lee, J.R., Sidiropoulos, A.: Genus and the geometry of the cut graph. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 193–201 (2010)Google Scholar
  42. 42.
    Leibon, G.: Random Delaunay Triangulations, the Thurston-Andreev Theorem and Metric Uniformization. PhD Thesis, University of California at San Diego (1999). Available on arXiv:math/0011016v1
  43. 43.
    Lévy, B., Mallet, J.-L.: Non-distorted texture mapping for sheared triangulated meshes. In: Proceedings of the 25th Annual Conference on Computer Graphics (SIGGRAPH), pp. 343–352 (1998)Google Scholar
  44. 44.
    Li, X., Xianfeng, G., Qin, H.: Surface mapping using consistent pants decomposition. IEEE Trans. Vis. Comput. Graph. 15, 558–571 (2009)CrossRefGoogle Scholar
  45. 45.
    Makover, E., McGowan, J.: The length of closed geodesics on random Riemann surfaces. Geom. Dedicata 151, 207–220 (2011)CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Matoušek, J., Sedgwick, E., Tancer, M., Wagner, U.: Untangling two systems of noncrossing curves. In: Wismath, S., Wolff, A. (eds.) Graph Drawing. Lecture Notes in Computer Science, vol. 8242, pp. 472–483. Springer, Cham (2013)Google Scholar
  47. 47.
    McKay, B.D., Wormald, N.C.: Automorphisms of random graphs with specified vertices. Combinatorica 4(4), 325–338 (1984)CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Piponi, D., Borshukov, G.: Seamless texture mapping of subdivision surfaces by model pelting and texture blending. In: Proceedings of the 27th Annual Conference on Computer Graphics (SIGGRAPH), pp. 471–478 (2000)Google Scholar
  49. 49.
    Pippenger, N., Schleich, K.: Topological characteristics of random triangulated surfaces. Random Struct. Algorithms 28(3), 247–288 (2006)CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Poon, S.-H., Thite, S.: Pants decomposition of the punctured plane. Preliminary version in Abstracts of the European Workshop on Computational Geometry (2006). arXiv:cs.CG/0602080
  51. 51.
    Przytycka, T.M., Przytycki, J.H.: On a lower bound for short noncontractible cycles in embedded graphs. SIAM J. Discrete Math. 3(2), 281–293 (1990)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Przytycka, T.M., Przytycki, J.H.: Surface triangulations without short noncontractible cycles. In: Robertson, N., Seymour, P. (eds.) Graph Structure Theory. Contemporary Mathematics, vol. 147. American Mathematical Society, Providence, RI (1993)Google Scholar
  53. 53.
    Przytycka, T.M., Przytycki, J.H.: A simple construction of high representativity triangulations. Discrete Math. 173, 209–228 (1997)CrossRefMATHMathSciNetGoogle Scholar
  54. 54.
    Pu, P.M.: Some inequalities in certain nonorientable riemannian manifolds. Pac. J. Math. 2, 55–71 (1952)CrossRefMATHGoogle Scholar
  55. 55.
    Read, R.C.: Some enumeration problems in graph theory. PhD Thesis, University of London (1958)Google Scholar
  56. 56.
    Robertson, N., Seymour, P.D.: Graph minors. VII. Disjoint paths on a surface. J. Comb. Theory Ser. B 45, 212–254 (1988)CrossRefMATHMathSciNetGoogle Scholar
  57. 57.
    Sabourau, S.: Asymptotic bounds for separating systoles on surfaces. Comment. Math. Helv. 83, 35–54 (2008)CrossRefMATHMathSciNetGoogle Scholar
  58. 58.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. II. Publish or Perish Press, Boston (1999)MATHGoogle Scholar
  59. 59.
    Stillwell, J.: Classical Topology and Combinatorial Group Theory. Springer, New York (1980)CrossRefMATHGoogle Scholar
  60. 60.
    Wood, Z., Hoppe, H., Desbrun, M., Schröder, P.: Removing excess topology from isosurfaces. ACM Trans. Graph. 23(2), 190–208 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Éric Colin de Verdière
    • 1
  • Alfredo Hubard
    • 2
  • Arnaud de Mesmay
    • 3
  1. 1.Département d’informatiqueÉcole normale supérieure, CNRSParisFrance
  2. 2.Laboratoire de l’Institut Gaspard MongeUniversité Paris-Est Marne-la-ValléeChamps-sur-MarneFrance
  3. 3.IST AustriaViennaAustria

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