Tree Checking for Sparse Complexes

  • Massimo Caboara
  • Sara Faridi
  • Peter Selinger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4151)


We detail here the sparse variant of the algorithm sketched in [2] for checking if a simplicial complex is a tree. A full worst case complexity analysis is given and several optimizations are discussed. The practical complexity is discussed for some examples.


Simplicial Complex Monomial Ideal Tree Decision Algorithm Relation Algorithm Facet Complex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Caboara, M., Faridi, S., Selinger, P.: Prototype implementation of tree algorithms, available at:
  2. 2.
    Caboara, M., Faridi, S., Selinger, P.: Simplicial cycles and the computation of simplicial trees. Journal of Symbolic Computation (to appear)Google Scholar
  3. 3.
    CoCoA Team, COCOA: a system for doing Computations in Commutative Algebra, available at:
  4. 4.
    Faridi, S.: The facet ideal of a simplicial complex. Manuscripta Mathematica 109, 159–174 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Faridi, S.: Cohen-Macaulay properties of square-free monomial ideals. Journal of Combinatorial Theory, Series A 109(2), 299–329 (2005)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Faridi, S.: Simplicial trees are sequentially Cohen-Macaulay. J. Pure and Applied Algebra 190, 121–136 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Faridi, S.: Monomial ideals via square-free monomial ideals. Lecture Notes in Pure and Applied Mathematics (to appear)Google Scholar
  8. 8.
    Simis, A., Vasconcelos, W., Villarreal, R.: On the ideal theory of graphs. J. Algebra 167(2), 389–416 (1994)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Villarreal, R.: Cohen-Macaulay graphs. Manuscripta Math. 66(3), 277–293 (1990)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Zheng, X.: Homological properties of monomial ideals associated to quasi-trees and lattices, Ph.D. thesis, Universität Duisburg-Essen (August 2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Massimo Caboara
    • 1
  • Sara Faridi
    • 2
  • Peter Selinger
    • 2
  1. 1.University of PisaItaly
  2. 2.Dalhousie UniversityHalifaxCanada

Personalised recommendations