Algorithmic Aspects of a General Modular Decomposition Theory

TL;DR

This paper introduces a broad modular decomposition framework inspired by graph theory, unifying various structures and extending classical concepts, while maintaining efficient algorithmic tools applicable across different domains.

Contribution

It presents a generalized modular decomposition theory applicable to multiple structures, extending classical graph modules and adapting existing algorithms for broader use.

Findings

Most algorithmic tools for graph modular decomposition apply to the general theory.

The generalized framework captures classical modules, 2-connected components, and star-cutsets.

An existing algorithm is extended to efficiently solve the new decomposition problem.

Abstract

A new general decomposition theory inspired from modular graph decomposition is presented. This helps unifying modular decomposition on different structures, including (but not restricted to) graphs. Moreover, even in the case of graphs, the terminology ``module'' not only captures the classical graph modules but also allows to handle 2-connected components, star-cutsets, and other vertex subsets. The main result is that most of the nice algorithmic tools developed for 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 1994 is generalised and yields a very efficient solution to the associated decomposition problem.