Combinatorica

, Volume 8, Issue 1, pp 91–102 | Cite as

Rubber bands, convex embeddings and graph connectivity

  • N. Linial
  • L. Lovász
  • A. Wigderson
Article

Abstract

We give various characterizations ofk-vertex connected graphs by geometric, algebraic, and “physical” properties. As an example, a graphG isk-connected if and only if, specifying anyk vertices ofG, the vertices ofG can be represented by points of ℝk−1 so that nok are on a hyper-plane and each vertex is in the convex hull of its neighbors, except for thek specified vertices. The proof of this theorem appeals to physics. The embedding is found by letting the edges of the graph behave like ideal springs and letting its vertices settle in equilibrium.

As an algorithmic application of our results we give probabilistic (Monte-Carlo and Las Vegas) algorithms for computing the connectivity of a graph. Our algorithms are faster than the best known (deterministic) connectivity algorithms for allk≧√n, and for very dense graphs the Monte Carlo algorithm is faster by a linear factor.

AMS subject classification (1980)

05 C 40 52 A 20 

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References

  1. [1]
    A. V.Aho J. E.Hopcropt and J. D.Ullman,The Design and Analysis of Computer Algorithms, Addison-Wesley, 1975.Google Scholar
  2. [2]
    S. Berkowitz, On computing the determinant in small parallel time using a small number of processors,Inform. Proc. Let.,18 (1984), 147–150.Google Scholar
  3. [3]
    G.Birkhoff and S.MacLane,A Survey of Modern Algebra, MacMillan, 1970.Google Scholar
  4. [4]
    D.Coppersmith and S.Winograd, On the asymptotic complexity of matrix multiplication,SIAM J. Computing, (1982), 472–492.Google Scholar
  5. [5]
    R. Connelly, Rigidity and Energy,Invent. Math.,66 (1982), 11–33.Google Scholar
  6. [6]
    S.Even,Graph Algorithms, Computer Science Press, 1979.Google Scholar
  7. [7]
    S. Even andR. E. Tarjan, Computing anst-numbering,Theoret. Comp. Sci.,2 (1976), 339–344.Google Scholar
  8. [8]
    Z. Galil, Finding the vertex connectivity of graphs,SIAM J. Computing,9 (1980), 197–199Google Scholar
  9. [9]
    A. W. Ingleton andM. J. Piff, Gammoids and transversal matroids,J. Comb. Theory,B15. (1973), 51–68.Google Scholar
  10. [10]
    L.Lovász, Combinatorial Problems and Exercises, North-Holland, 1979.Google Scholar
  11. [11]
    A.Lempel, S.Even and I.Cederbaum, An algorithm for planarity testing of graphs,Theory of Graphs, (1967),International Symposium, Rome, P. Rosensfield ed., 215–232.Google Scholar
  12. [12]
    H. Perfect, Symmetrized form of P. Hall's theorem on distinct representatives,Quart. J. Math. Oxford,17 (1966), 303–306.Google Scholar
  13. [13]
    J. T. Schwartz, Fast probabilistic algorithms for verification of polynomial identities,J. ACM 27,4 (1980), 701–717.Google Scholar
  14. [14]
    V.Vishkin, Synchronous parallel computation, a survey,TR#71,Department of Computer Science, Courant Institute, NYU, 1983.Google Scholar
  15. [15]
    W. T. Tutte, How to draw a graph,Proc. London Math. Soc.,13 (1963), 743–768.Google Scholar

Copyright information

© Akadémiai Kiadó 1988

Authors and Affiliations

  • N. Linial
    • 1
    • 3
  • L. Lovász
    • 1
    • 2
  • A. Wigderson
    • 1
    • 3
  1. 1.Mathematical Sciences Research InstituteBerkeleyUSA
  2. 2.Eötvös Loránd UniversityBudapest
  3. 3.The Hebrew UniversityJerusalemIsrael

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