Computing the directed Cartesian-product decomposition of a directed graph from its undirected decomposition in linear time

In this paper, we design an algorithm that, given a directed graph GG and the Cartesian-product decomposition of its underlying undirected graph G̃, produces the directed Cartesian-product decomposition of GG in linear time. This is the first time that the linear complexity is achieved for this problem, which has two major consequences. Firstly, it shows that the directed and undirected versions of the Cartesian-product decomposition of graphs are linear-time equivalent problems. And secondly, as there already exists a linear-time algorithm for solving the undirected version of the problem, combined together, it provides the first linear-time algorithm for computing the directed Cartesian-product decomposition of a directed graph.