  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.
