On the Strongly Connected and Biconnected Components of the Complement of Graphs

In this paper, we consider the problems of computing the strongly connected components and the biconnected components of the complement of a given graph. In particular, for a directed graph G on n vertices and m edges, we present a simple algorithm for computing the strongly connected components of G ¯ which runs in optimal O ( n + m ) time. The algorithm can be parallelized to yield an O ( log 2 n ) -time and O ( m 1.188 / log n ) -processor solution. As a byproduct, we obtain a very simple optimal parallel co-connectivity algorithm. onally, we establish properties which, for an undirected graph on n vertices and m edges, enable us to describe an O ( n + m ) -time algorithm for computing the biconnected components of G ¯ , which can be parallelized resulting in an algorithm that runs in O ( log n ) time using O ( ( n + m ) / log n ) processors.