Rubber bands, convex embeddings and graph connectivity

N. Linial, L. Lovász, A. Wigderson

Research output: Contribution to journalArticle

42 Citations (Scopus)

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 languageEnglish
Pages (from-to)91-102
Number of pages12
JournalCombinatorica
Volume8
Issue number1
DOIs
Publication statusPublished - Mar 1988

Fingerprint

Graph Connectivity
Rubber
Graph in graph theory
Connectivity
Monte Carlo Algorithm
Appeal
Vertex of a graph
Physical property
Convex Hull
Connected graph
Physics
Physical properties
If and only if
Computing
Theorem

Keywords

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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Mathematics(all)

Cite this

Rubber bands, convex embeddings and graph connectivity. / Linial, N.; Lovász, L.; Wigderson, A.

In: Combinatorica, Vol. 8, No. 1, 03.1988, p. 91-102.

Research output: Contribution to journalArticle

Linial, N. ; Lovász, L. ; Wigderson, A. / Rubber bands, convex embeddings and graph connectivity. In: Combinatorica. 1988 ; Vol. 8, No. 1. pp. 91-102.
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