# Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

## 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## Notes

### Acknowledgments

We would like to thank Jean-Daniel Boissonnat, Ramsay Dyer, and Arijit Ghosh for pointing out and discussing with us their results on Voronoi diagrams of Riemannian surfaces [16] and manifolds. We are grateful to the anonymous referees for their careful reading of the manuscript, which allowed to correct several problems and to improve the presentation significantly, and for pointing out Kowalick’s thesis [38]. This work was supported by the French ANR Blanc Project ANR-12-BS02-005 (RDAM). Portions of this work were done during a post-doctoral visit of the second author at the Département d’informatique of École normale supérieure, funded by the Fondation Sciences Mathématiques de Paris. Portions of this work were done while the third author was a Ph.D. student at the Département d’informatique of École normale supérieure.

## References

- 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.Boissonnat, J.-D., Dyer, R., Ghosh, A.: Delaunay triangulations of manifolds. arXiv:1311.0117 (2013)
- 3.Brooks, R., Makover, E.: Random construction of Riemann surfaces. J. Differ. Geom.
**68**(1), 121–157 (2004)MATHMathSciNetGoogle Scholar - 4.Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Progress in Mathematics, vol. 106. Birkhäuser, Basel (1992)Google Scholar
- 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.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.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.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.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.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.Chambers, E.W., Erickson, J., Nayyeri, A.: Homology flows, cohomology cuts. SIAM J. Comput.
**41**(6), 1605–1634 (2012)CrossRefMATHMathSciNetGoogle Scholar - 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.Colin de Verdière, É., Erickson, J.: Tightening nonsimple paths and cycles on surfaces. SIAM J. Comput.
**39**(8), 3784–3813 (2010)CrossRefMATHMathSciNetGoogle Scholar - 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.do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)CrossRefMATHGoogle Scholar
- 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.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.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.Erickson, J., Har-Peled, S.: Optimally cutting a surface into a disk. Discrete Comput. Geom.
**31**(1), 37–59 (2004)CrossRefMATHMathSciNetGoogle Scholar - 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.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.Erickson, J., Worah, P.: Computing the shortest essential cycle. Discrete Comput. Geom.
**44**(4), 912–930 (2010)CrossRefMATHMathSciNetGoogle Scholar - 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.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.Geelen, J., Huynh, T., Richter, R.B.: Explicit bounds for graph minors. arxiv:1305.1451 (2013)
- 26.Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom.
**18**, 1–147 (1983)MATHMathSciNetGoogle Scholar - 27.Gromov, M.: Systoles and intersystolic inequalities. In: Actes de la table ronde de géométrie différentielle, pp. 291–362 (1992)Google Scholar
- 28.Gromov, M.: Sign and geometric meaning of curvature. Rend. Semin. Mat. Fis. Milano
**61**(1991), 9–123 (1994)MathSciNetGoogle Scholar - 29.Guskov, I., Wood, Z.J.: Topological noise removal. In: Proceedings of Graphics Interface, pp. 19–26 (2001)Google Scholar
- 30.Guth, L., Parlier, H., Young, R.: Pants decompositions of random surfaces. Geom. Funct. Anal.
**21**, 1069–1090 (2011)CrossRefMATHMathSciNetGoogle Scholar - 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.Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002). Available at http://www.math.cornell.edu/hatcher/
- 33.Hutchinson, J.P.: On short noncontractible cycles in embedded graphs. SIAM J. Discrete Math.
**1**(2), 185–192 (1988)CrossRefMATHMathSciNetGoogle Scholar - 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.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.Kervaire, M.A.: A manifold which does not admit any differentiable structure. Comment. Math. Helv.
**34**, 257–270 (1960)CrossRefMATHMathSciNetGoogle Scholar - 37.Klingenberg, W.: Riemannian Geometry. Philosophie und Wissenschaft. de Gruyter, Berlin (1995)CrossRefGoogle Scholar
- 38.Kowalick, R.: Discrete systolic inequalities. PhD Thesis, Ohio State University (2013). http://rave.ohiolink.edu/etdc/view?acc_num=osu1384873457
- 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.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.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.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.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.Li, X., Xianfeng, G., Qin, H.: Surface mapping using consistent pants decomposition. IEEE Trans. Vis. Comput. Graph.
**15**, 558–571 (2009)CrossRefGoogle Scholar - 45.Makover, E., McGowan, J.: The length of closed geodesics on random Riemann surfaces. Geom. Dedicata
**151**, 207–220 (2011)CrossRefMATHMathSciNetGoogle Scholar - 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.McKay, B.D., Wormald, N.C.: Automorphisms of random graphs with specified vertices. Combinatorica
**4**(4), 325–338 (1984)CrossRefMATHMathSciNetGoogle Scholar - 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.Pippenger, N., Schleich, K.: Topological characteristics of random triangulated surfaces. Random Struct. Algorithms
**28**(3), 247–288 (2006)CrossRefMATHMathSciNetGoogle Scholar - 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.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.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.Przytycka, T.M., Przytycki, J.H.: A simple construction of high representativity triangulations. Discrete Math.
**173**, 209–228 (1997)CrossRefMATHMathSciNetGoogle Scholar - 54.Pu, P.M.: Some inequalities in certain nonorientable riemannian manifolds. Pac. J. Math.
**2**, 55–71 (1952)CrossRefMATHGoogle Scholar - 55.Read, R.C.: Some enumeration problems in graph theory. PhD Thesis, University of London (1958)Google Scholar
- 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.Sabourau, S.: Asymptotic bounds for separating systoles on surfaces. Comment. Math. Helv.
**83**, 35–54 (2008)CrossRefMATHMathSciNetGoogle Scholar - 58.Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. II. Publish or Perish Press, Boston (1999)MATHGoogle Scholar
- 59.Stillwell, J.: Classical Topology and Combinatorial Group Theory. Springer, New York (1980)CrossRefMATHGoogle Scholar
- 60.Wood, Z., Hoppe, H., Desbrun, M., Schröder, P.: Removing excess topology from isosurfaces. ACM Trans. Graph.
**23**(2), 190–208 (2004)CrossRefGoogle Scholar