We consider the problem of finding a subcomplex $\mathcal{K}'$ of a simplicial complex $\mathcal{K}$ such that $\mathcal{K}'$ is homeomorphic to the 2-dimensional sphere, $\mathbb{S}^2$. We study two variants of this problem. The first asks if there exists such a $\mathcal{K}'$ with at most $\mathcal{K}$ triangles, and we show that this variant is ${\mathsf{W[1]}}$-hard and, assuming the exponential time hypothesis, admits no $n^{o(\sqrt{k})}$-time algorithm. We also give an algorithm that is tight with regard to this lower bound. The second problem is the dual of the first and asks if $\mathcal{K}'$ can be found by removing at most $k$ triangles from $\mathcal{K}$. This variant has an immediate $\mathcal{O}(3^{k}poly(|\mathcal{K}|))$-time algorithm, and we show that it admits a polynomial kernelization to $\mathcal{O}(k^2)$ triangles, as well as a polynomial compression to a weighted version with bit-size $\mathcal{O}(k \log k)$.


  1. computational topology
  2. parameterized complexity
  3. simplicial complex

MSC codes

  1. 57M20
  2. 68Q17
  3. 68Q25

Get full access to this article

View all available purchase options and get full access to this article.


N. Alon, R. Yuster, and U. Zwick, Color-coding, J. ACM, 42 (1995), pp. 844--856, https://doi.org/10.1145/210332.210337.
B. Bagchi, B. A. Burton, B. Datta, N. Singh, and J. Spreer, Efficient algorithms to decide tightness, in 32nd International Symposium on Computational Geometry, S. Fekete and A. Lubiw, eds., LIPIcs. Leibniz Int. Proc. Inform. 51, Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2016, pp. 12:1--12:15, https://doi.org/10.4230/LIPIcs.SoCG.2016.12.
N. Bonichon, C. Gavoille, N. Hanusse, D. Poulalhon, and G. Schaeffer, Planar graphs, via well-orderly maps and trees, Graphs Combin., 22 (2006), pp. 185--202, https://doi.org/10.1007/s00373-006-0647-2.
A. Boulch, É. Colin de Verdière, and A. Nakamoto, Irreducible triangulations of surfaces with boundary, Graphs Combin., 29 (2013), pp. 1675--1688, https://doi.org/10.1007/s00373-012-1244-1.
B. A. Burton and R. G. Downey, Courcelle's theorem for triangulations, J. Combin. Theory Ser. A, 146 (2016), pp. 264--294, https://doi.org/10.1016/j.jcta.2016.10.001.
B. A. Burton, C. Maria, and J. Spreer, Algorithms and complexity for Turaev-Viro invariants, in Automata, Languages, and Programming: 42nd International Colloquium, Part 1, Springer, New York, 2015, pp. 281--293, https://doi.org/10.1007/978-3-662-47672-7_23.
B. A. Burton and W. Pettersson, Fixed parameter tractable algorithms in combinatorial topology, in Computing and Combinatorics: 20th International Conference, Z. Cai, A. Zelikovsky, and A. Bourgeois, eds., Springer, New York, 2014, pp. 300--311, https://doi.org/10.1007/978-3-319-08783-2_26.
O. Busaryev, S. Cabello, C. Chen, T. K. Dey, and Y. Wang, Annotating simplices with a homology basis and its applications, in Algorithm Theory, 13th Scandinavian Symposium and Workshops, F. V. Fomin and P. Kaski, eds., Lecture Notes in Comput. Sci. 7357, Springer, New York, 2012, pp. 189--200, https://doi.org/10.1007/978-3-642-31155-0_17.
C. Chen and D. Freedman, Hardness results for homology localization, Discrete Comput. Geom., 45 (2011), pp. 425--448, https://doi.org/10.1007/s00454-010-9322-8.
M. Cygan, F. V. Fomin, Ł. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, and M. P. and S. Saurabh, Parameterized Algorithms, Springer, New York, 2015, https://doi.org/10.1007/978-3-319-21275-3.
R. G. Downey and M. R. Fellows, Fundamentals of Parameterized Complexity, Texts Comput. Sci., Springer, New York, 2013, https://doi.org/10.1007/978-1-4471-5559-1.
J. Erickson and A. Nayyeri, Minimum cuts and shortest non-separating cycles via homology covers, in Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, D. Randall, ed., SIAM, Philadelphia, 2011, pp. 1166--1176, https://doi.org/10.1137/1.9781611973082.88.
M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom., 17 (1982), pp. 357--453, http://projecteuclid.org/euclid.jdg/1214437136.
Z. Gao, The number of rooted triangular maps on a surface, J. Combin. Theory Ser. B, 52 (1991), pp. 236--249, https://doi.org/10.1016/0095-8956(91)90065-R.
J. R. Gilbert, J. P. Hutchinson, and R. E. Tarjan, A separator theorem for graphs of bounded genus, J. Algorithms, 5 (1984), pp. 391--407, https://doi.org/10.1016/0196-6774(84)90019-1.
J. Hass and G. Kuperberg, The complexity of recognizing the 3-sphere, Oberwolfach Rep., 24 (2012), pp. 1425--1426, https://doi.org/10.4171/owr/2012/24.
S. Ivanov, Computational Complexity (Answer). MathOverflow, http://mathoverflow.net/questions/118357/computational-complexity (2016).
K. Kawarabayashi, B. Mohar, and B. A. Reed, A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded tree-width, in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, IEEE, New York, 2008, pp. 771--780, https://doi.org/10.1109/FOCS.2008.53.
C. Maria and J. Spreer, A polynomial time algorithm to compute quantum invariants of 3-manifolds with bounded first Betti number, in Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, P. N. Klein, ed., SIAM, Philadelphia, 2017, pp. 2721--2732, https://doi.org/10.1137/1.9781611974782.180.
D. Marx, A tight lower bound for planar multiway cut with fixed number of terminals, in Automata, Languages, and Programming--39th International Colloquium, Part I, A. Czumaj, K. Mehlhorn, A. M. Pitts, and R. Wattenhofer, eds., Lecture Notes in Comput. Sci. 7391, Springer, New York, 2012, pp. 677--688, https://doi.org/10.1007/978-3-642-31594-7_57.
J. Matou\vsek, Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry, Universitext, Springer, New York, 2007.
B. Mohar, A linear time algorithm for embedding graphs in an arbitrary surface, SIAM J. Discrete Math., 12 (1999), pp. 6--26, https://doi.org/10.1137/S089548019529248X.
J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley, Reading, MA, 1993.
A. Nabutovsky, Einstein structures: Existence versus uniqueness, Geom. Funct. Anal., 5 (1995), pp. 76--91, https://doi.org/10.1007/BF01928216.
S. Schleimer, Sphere recognition lies in NP, in Low-Dimensional and Symplectic Topology, M. Usher, ed., Proc. Sympos. Pure Math. 82, AMS, Providence, RI, 2011, pp. 183--213.
W. Tutte, A census of planar triangulations, Canad. J. Math., 14 (1962), pp. 21--38, https://doi.org/10.4153/CJM-1962-002-9.

Information & Authors


Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 2092 - 2110
ISSN (online): 1095-7146


Submitted: 2 February 2018
Accepted: 31 July 2019
Published online: 29 October 2019


  1. computational topology
  2. parameterized complexity
  3. simplicial complex

MSC codes

  1. 57M20
  2. 68Q17
  3. 68Q25



Funding Information

Javna Agencija za Raziskovalno Dejavnost RS https://doi.org/10.13039/501100004329 : P1–0297, L7–5459, J1–8130

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.