Determining Whether a Simplicial 3-Complex Collapses to a 1-Complex Is NP-Complete

  • Rémy Malgouyres
  • Angel R. Francés
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4992)


We show that determining whether or not a simplicial 2 − complex collapses to a point is deterministic polynomial time decidable. We do this by solving the problem of constructively deciding whether a simplicial 2 −complex collapses to a 1 −complex. We show that this proof cannot be extended to the 3D case, by proving that deciding whether a simplicial 3 −complex collapses to a 1 −complex is an NP −complete problem.


Simplicial Topology Collapsing Computational Complexity NP −completeness 


  1. [B99]
    Bertrand, G.: New Notions for Discrete Topology. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, Springer, Heidelberg (1999)Google Scholar
  2. [B64]
    Bing, R.H.: Some aspects of the topology of 3 −manifolds related to the Poincaré conjecture. In: Saaty, T.L. (ed.) Lectures on Modern Mathematics, vol. II, pp. 93–128. Wiley, Chichester (1964)Google Scholar
  3. [C71]
    Cook, S.A.: The complexity of Theorem Proving Procedures. In: Proc. 3rd Ann. ACM Symp. on Theory of Computing, Association for Computing Machinery, New-York, pp. 151–158Google Scholar
  4. [EG96]
    Egecioglu, O., Gonzalez, T.F.: A Computationally Intractable Problem on Simplicial Complexes. Computational Geometry, Theory and Applications 6, 85–98 (1996)zbMATHMathSciNetGoogle Scholar
  5. [F00]
    Fourey, S., Malgouyres, R.: A concise characterization of 3D simple points. Discrete Applied Mathematics 125(1), 59–80 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [GJ79]
    Garey, M.R., Johnson, D.S.: Computers and Intractability: a guide to the theory of NP −completeness. W.H. Freeman and Company publishers, New YorkGoogle Scholar
  7. [K97]
    Kong, T.Y.: Topology-Preserving Deletion of 1’s from 2 −, 3 − and 4 − Dimensional Binary Images. In: Ahronovitz, E. (ed.) DGCI 1997. LNCS, vol. 1347, pp. 3–18. Springer, Heidelberg (1997)Google Scholar
  8. [KR01]
    Kong, T.Y., Roscoe, A.W.: Simple Points in 4 −dimensional (and Higher-Dimensional) Binary Images (paper in preparation)Google Scholar
  9. [VL90]
    : Handbook of theoretical computer science. In: Van Leeuwen, J. (ed.) Algorithms and complexity, vol. A, pp. 67–161. Elsevier Science Publishers, Amsterdam (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Rémy Malgouyres
    • 1
  • Angel R. Francés
    • 2
  1. 1.Laboratoire d’Algorithmique et Image (LAIC, EA2146), IUT département informatiqueUniv. Clermont 1Aubière cedexFrance
  2. 2.Dpto. Informática e Ingeniería de Sistemas, Facultad de CienciasUniversidad de ZaragozaZaragozaSpain

Personalised recommendations