I have to solve an independent set problem (ISP) on dense graphs with many cliques. To tackle the problem, I'm considering to use graph decompositions such as tree-, modular decomposition or clique-width k-expression trees.
Are there some decompositions (or other techniques) favorable for my type of graphs?
From what I understand, considering e.g. the tree-decomposition, the ISP can be solved in $\mathcal{O}(n2^k)$ where $n$ is number of vertices of the tree and $k$ the tree-width. Moreover, the tree-width is a measure of how similar a graph is to a tree. From this, I concluded that a graph which is not very similar to tree (maybe just like mine) has a large tree-width and, hence, the tree-decomposition is maybe not a favorable decomposition to solve the ISP.
Is e.g. the clique-width k-expression tree more suitable for my case?