On computing the $2$-vertex-connected components of directed graphs

In this paper we consider the problem of computing the 2-vertex-connected components (2-vccs) of directed graphs. We present two new algorithms for solving this problem. The first algorithm runs in O(mn) time, the second in O(nm) time. Furthermore, we show that the old algorithm of Erusalimskii and Svetlov runs inO(nm) time. In this paper, we investigate the relationship between 2-vccs and dominator trees. We also present an algorithm for computing the 3-vertex-connected components (3-vccs) of a directed graph in O(nm) time, and we show that the k-vertexconnected components (k-vccs) of a directed graph can be computed in O(mn) time. Finally, we consider three applications of our new algorithms, which are approximation algorithms for problems that are generalization of the problem of approximating the smallest 2-vertex-connected spanning subgraph of 2-vertex-connected directed graph.