Property Testing on k-Vertex-Connectivity of Graphs

Abstract

We present an algorithm for testing the k-vertex-connectivity of graphs with given maximum degree. The time complexity of the algorithm is independent of the number of vertices and edges of graphs. A graph G with n vertices and 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 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 ${O\left(d\left(\frac{c}{\epsilon d}\right)^{k}\log\frac{1}{\epsilon d}\right)}$ time (c > 1 is a constant) for given (k − 1)-vertex-connected graphs, and ${O\left(d\left(\frac{ck}{\epsilon d}\right)^{k}\log\frac{k}{\epsilon d}\right)}$ time (c > 1 is a constant) for given general graphs. It is the first constant-time k-vertex-connectivity testing algorithm for general k ≥ 4.