Graph decompositions and factorizing permutations

A factorizing permutation of a given undirected graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. We apply it to modular decomposition and we propose a linear algorithm that computes the whole decomposition tree when a factorizing permutation is provided. This algorithm can be seen as a common generalization of (Ma and Hsu, 1991) for modular decomposition of chordal graphs and (Habib et al., 1995) for inheritance graph decomposition. It also suggests many new decomposition algorithms for various notions of graph decompositions.