Efficient Connectivity Testing of Hypercubic Networks with Faults

  • Tomáš Dvořák
  • Jiří Fink
  • Petr Gregor
  • Václav Koubek
  • Tomasz Radzik
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)


Given a connected graph G and a set F of faulty vertices of G, let G − F be the graph obtained from G by deletion of all vertices of F and edges incident with them. Is there an algorithm, whose running time may be bounded by a polynomial function of |F| and log|V(G)|, which decides whether G − F is still connected? Even though the answer to this question is negative in general, we describe an algorithm which resolves this problem for the n-dimensional hypercube in time O(|F|n3). Furthermore, we sketch a more general algorithm that is efficient for graph classes with good vertex expansion properties.


Connected Graph Interconnection Network Graph Class Local Connectivity Isoperimetric Problem 
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.


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  1. 1.
    Chan, M.Y., Lee, S.-J.: On the existence of Hamiltonian circuits in faulty hypercubes. SIAM J. Discrete Math. 4, 511–527 (1991)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, Y.-C., Huang, Y.-Z., Hsu, L.-H., Tan, J.J.M.: A family of Hamiltonian and Hamiltonian connected graphs with fault tolerance. J. Supercomput. 54, 229–238 (2010)CrossRefGoogle Scholar
  3. 3.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  4. 4.
    Dvořák, T., Gregor, P.: Partitions of faulty hypercubes into paths with prescribed endvertices. SIAM J. Discrete Math. 22, 1448–1461 (2008)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dvořák, T., Koubek, V.: Long paths in hypercubes with a quadratic number of faults. Inf. Sci. 179, 3763–3771 (2009)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dvořák, T., Koubek, V.: Computational complexity of long paths and cycles in faulty hypercubes. Theor. Comput. Sci. 411, 3774–3786 (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Fink, J., Gregor, P.: Long paths and cycles in hypercubes with faulty vertices. Inf. Sci. 179, 3634–3644 (2009)CrossRefzbMATHGoogle Scholar
  8. 8.
    Harper, L.H.: Optimal Numberings and Isoperimetric Problems on Graphs. J. Comb. Theory 1, 385–393 (1966)CrossRefzbMATHGoogle Scholar
  9. 9.
    Knuth, D.E.: The Art of Computer Programming: Sorting and Searching, 2nd edn., vol. III. Addison-Wesley, Reading (1998)zbMATHGoogle Scholar
  10. 10.
    Leighton, F.T.: Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann, San Mateo (1992)zbMATHGoogle Scholar
  11. 11.
    Park, J.-H., Kim, H.-C., Lim, H.-S.: Many-to-Many Disjoint Path Covers in the Presence of Faulty Elements. IEEE Trans. Comput. 58, 528–540 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Tomáš Dvořák
    • 1
  • Jiří Fink
    • 1
  • Petr Gregor
    • 1
  • Václav Koubek
    • 1
  • Tomasz Radzik
    • 2
  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  2. 2.Department of Computer ScienceKing’s College LondonUnited Kingdom

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