# Modular Decomposition and Clique-width

I am trying to understand some concepts about Modular decomposition and Clique-width graphs.

In this paper ("On P4-tidy graphs"), there is a proof of how to solve optimization problems like clique-number or chromatic-number using Modular decomposition. Solving these problems by composing (using disjoint sum or disjoint union) two graphs G1,G2 is easy when you know the answer for G1 and G2. Since the prime-graphs on the decomposition of P4-tidy graphs are bounded graphs (i.e. C5,P5,etc.), it is easy to solve it for these "base case" and then solve it for compositions. Hence by using the decomposition tree it is possible to solve these problems in linear time.

But it seems that this technique would work with any graph class such that graph-primes are bounded. Then I found this paper "Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width" which seems to make the generalization I was looking for but I couldn't understand it very well.

My question are:

1- Is equivalent to say that prime-graphs of the decomposition tree are bounded (like in the P4-tidy graphs case) and say that a graph has the property "Clique-Width" bounded?

2- In case the answer for 1 is NO, then: Does exists any result about classes of graphs with graph-primes bounded (like in P4-tidy graphs) and thus optimization problems like clique-number solvable on linear time on all these classes?