Property Testing on k-Vertex-Connectivity of Graphs

  • Authors:

  • Yuichi Yoshida;Hiro Ito

  • Affiliations:

  • School of Informatics, Kyoto University, Kyoto, Japan 606-8501;School of Informatics, Kyoto University, Kyoto, Japan 606-8501

  • Venue:
  • ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
  • Year:
  • 2008

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Abstract

We present an algorithm for testing thek-vertex-connectivity of graphs with given maximum degree.The time complexity of the algorithm is independent of the numberof vertices and edges of graphs. A graph G with nvertices and maximum degree at most d is calledε-far from k-vertex-connectivity when atleast $\frac{\epsilon dn}{2}$ edges must be added to or removedfrom G to obtain a k-vertex-connected graph withmaximum degree at most d. The algorithm always acceptsevery graph that is k-vertex-connected and rejects everygraph that is ε-far fromk-vertex-connectivity with a probability of at least 2/3.The algorithm runs in ${O\left(d\left(\frac{c}{\epsilond}\right)^{k}\log\frac{1}{\epsilon d}\right)}$ time (c 1 is a constant) for given (k - 1)-vertex-connectedgraphs, and ${O\left(d\left(\frac{ck}{\epsilond}\right)^{k}\log\frac{k}{\epsilon d}\right)}$ time (c 1 is a constant) for given general graphs. It is the firstconstant-time k-vertex-connectivity testing algorithm forgeneral k ≥ 4.