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The three standard products (the Cartesian, the direct and the strongproduct) of undirected graphs have been wellinvestigated, unique prime factordecomposition (PFD) are known and polynomial time algorithms have beenestablished for determining the prime factors.  For directed graphs, unique PFD results with respect to the standard productsare known. However, there is still a lack of algorithms, that computes the PFDof directed graphs with respect to the direct and the strong product ingeneral. In this contribution, we focus on the algorithmic aspects fordetermining the PFD of directed graphs with respect to the strong product.Essential for computing the prime factors is the construction of a so-calledCartesian skeleton. This article introduces the notion of the Cartesianskeleton of directed graphs as a generalization of the Cartesian skeleton ofundirected graphs. We provide new, fast and transparent algorithms for itsconstruction. Moreover, we present a first polynomial time algorithm fordetermining the PFD with respect to the strong product of arbitrary connecteddigraphs.

On the Cartesian Skeleton and the Factorization of the Strong Product of Digraphs