Rubber bands, convex embeddings and graph connectivity

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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    A. V.Aho J. E.Hopcropt and J. D.Ullman,The Design and Analysis of Computer Algorithms, Addison-Wesley, 1975.

  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.

  4. [4]

    D.Coppersmith and S.Winograd, On the asymptotic complexity of matrix multiplication,SIAM J. Computing, (1982), 472–492.

  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.

  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), 197–199

    Google 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.

  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.

  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.

  15. [15]

    W. T. Tutte, How to draw a graph,Proc. London Math. Soc.,13 (1963), 743–768.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Linial, N., Lovász, L. & Wigderson, A. Rubber bands, convex embeddings and graph connectivity. Combinatorica 8, 91–102 (1988). https://doi.org/10.1007/BF02122557

Download citation

AMS subject classification (1980)

  • 05 C 40
  • 52 A 20