[math]-Modules in Graphs

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 566-589, March 2024.
Abstract. Modular decomposition focuses on repeatedly identifying a module [math] (a collection of vertices that shares exactly the same neighborhood outside of [math]) and collapsing it into a single vertex. This notion of exactitude of neighborhood 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 [math]-module, a relaxation that 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 [math]-modules can be computed in polynomial time, and we generalize series and parallel operation between graphs. This leads to [math]-cographs which have interesting properties. We study how to generalize Gallai’s theorem corresponding to the case for [math], but unfortunately we give evidence that computing such a decomposition tree can be difficult.