({\alpha}, {\beta})-Modules in Graphs

TL;DR

This paper introduces ({}, {})-modules as a relaxed form of modular decomposition in graphs, allowing bounded errors while preserving algebraic properties, and demonstrates their polynomial-time computability and existence of a decomposition tree.

Contribution

It generalizes modular decomposition by defining ({}, {})-modules, maintaining algebraic structure and polynomial-time computability, extending Gallai's Theorem.

Findings

Minimal ({}, {})-modules can be computed in polynomial time.

Every graph admits an ({},{})-modular decomposition tree.

Computing the decomposition tree may be computationally hard.

Abstract

Modular Decomposition focuses on repeatedly identifying a module M (a collection of vertices that shares exactly the same neighbourhood outside of M) and collapsing it into a single vertex. This notion of exactitude of neighbourhood is very strict, especially when dealing with real world graphs. We study new ways to relax this exactitude condition. However, generalizing modular decomposition is far from obvious. Most of the previous proposals lose algebraic properties of modules and thus most of the nice algorithmic consequences. We introduce the notion of an ({\alpha}, {\beta})-module, a relaxation that allows a bounded number of errors in each node and maintains some of the algebraic structure. It leads to a new combinatorial decomposition with interesting properties. Among the main results in this work, we show that minimal ({\alpha}, {\beta})-modules can be computed in polynomial time, and that every graph admits an ({\alpha},{\beta})-modular decomposition tree, thus generalizing Gallai's Theorem (which corresponds to the case for {\alpha} = {\beta} = 0). Unfortunately we give evidence that computing such a decomposition tree can be difficult.