Algorithmic aspects of a general modular decomposition theory

A new general decomposition theory inspired by modular graph decomposition is presented. This helps in unifying modular decomposition on different structures, including (but not restricted to) graphs. Moreover, even in the case of graphs, this new notion called homogeneous modules not only captures the classical graph modules but also allows handling 2-connected components, star-cutsets, and other vertex subsets.The main result is that most of the nice algorithmic tools developed for the modular decomposition of graphs still apply efficiently on our generalisation of modules. Besides, when an essential axiom is satisfied, almost all the important properties can be retrieved. For this case, an algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan [A. Ehrenfeucht, H. Gabow, R. McConnell, S. Sullivan, An O(n2)O(n2) Divide-and-Conquer Algorithm for the prime tree decomposition of two-structures and modular decomposition of graphs, Journal of Algorithms 16 (1994) 283–294.] is generalised and yields a very efficient solution to the associated decomposition problem.