Abstract
We present an algorithm for testing the k-vertex-connectivity of graphs with the given maximum degree. The time complexity of the algorithm is independent of the number of vertices and edges of graphs. Fixed degree bound d, a graph G with n vertices and a maximum degree at most d is called ε-far from k-vertex-connectivity when at least \(\frac{\epsilon dn}{2}\) edges must be added to or removed from G to obtain a k-vertex-connected graph with a maximum degree at most d. The algorithm always accepts every graph that is k-vertex-connected and rejects every graph that is ε-far from k-vertex-connectivity with a probability of at least 2/3. The algorithm runs in \(O(d(\frac{c}{\epsilon d})^{k}\log\frac {1}{\epsilon d})\) time (c>1 is a constant) for (k−1)-vertex-connected graphs, and in \(O(d(\frac{ck}{\epsilon d})^{k}\log\frac{k}{\epsilon d})\) time (c>1 is a constant) for general graphs. It is the first constant-time k-vertex-connectivity testing algorithm for general k≥4.
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A preliminary version of this paper appeared in Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP’08), #LNCS, vol. 5125, pp. 539–550, 2008.
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Yoshida, Y., Ito, H. Property Testing on k-Vertex-Connectivity of Graphs. Algorithmica 62, 701–712 (2012). https://doi.org/10.1007/s00453-010-9477-y
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DOI: https://doi.org/10.1007/s00453-010-9477-y