A Linear-Time Algorithm for Computing the Prime Decomposition of a Directed Graph with Regard to the Cartesian Product

In this paper, we design the first linear-time algorithm for computing the prime decomposition of a digraph G with regard to the cartesian product. A remarkable feature of our solution is that it computes the decomposition of G from the decomposition of its underlying undirected graph, for which there exists a linear-time algorithm. First, this allows our algorithm to remain conceptually very simple and in addition, it provides new insight into the connexions between the directed and undirected versions of cartesian product of graphs. The general idea of graph decompositions is to describe a graph as the composition, through some operations, of a set of simpler (and usually smaller) graphs. This framework has turned out to be very useful both for proving theorems (see e.g. [2]) and for solving efficiently difficult algorithmic problems using the ”divide and conquer” approach (see [10]). This is the reason why, in the last decades, a lot of effort have been made for computing efficiently the decomposition of a graph with respect to a given operation. The cartesian product of undirected graphs (graphs for short) and directed graphs (digraphs for short), usually denoted by , is a classical and useful decomposition operation that allows to factorise some specifically structured redundancy in a graph. Such redundancy naturally appears in various contexts, both theoretic and practical, such as accountability, databases or programming. In those contexts, revealing and factorising these redundancy has a great impact on the efficiency of the solutions proposed to manage these systems. Cartesian product has been used from the early times of graph theory, but the first intensive studies were provided by Sabidussi [9] and Vizing [11]. ∗This work was partially supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM). †Université Claude Bernard Lyon 1, DANTE/INRIA, LIP UMR CNRS 5668, ENS de Lyon, Université de Lyon, [email protected] ‡ENS de Lyon, LIP UMR CNRS 5668, Université de Lyon, [email protected] §ENS de Lyon, Université de Lyon, [email protected]