Rubber bands, convex embeddings and graph connectivity

N. Linial*, L. Lovász, A. Wigderson

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

60 Scopus citations

Abstract

We give various characterizations of k-vertex connected graphs by geometric, algebraic, and "physical" properties. As an example, a graph G is k-connected if and only if, specifying any k vertices of G, the vertices of G can be represented by points of ℝk-1 so that no k are on a hyper-plane and each vertex is in the convex hull of its neighbors, except for the k 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 all k≧√n, and for very dense graphs the Monte Carlo algorithm is faster by a linear factor.

Original languageAmerican English
Pages (from-to)91-102
Number of pages12
JournalCombinatorica
Volume8
Issue number1
DOIs
StatePublished - Mar 1988

Keywords

  • AMS subject classification (1980): 05C40, 52A20

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