On clique separators, nearly chordal graphs, and the maximum weight stable set problem

Clique separators in graphs are a helpful tool used by Tarjan as a divide-and-conquer approach for solving various graph problems such as the Maximum Weight Stable Set (MWS) Problem, Coloring and Minimum Fill-in but few examples are known where this approach was used. We combine decomposition by clique separators and by homogeneous sets and show that the resulting binary tree gives an efficient way for solving the MWS problem. Moreover, we combine this approach with the concept of nearly chordal and nearly perfect and obtain some new graph classes where MWS is solvable in polynomial time by our approach. On some of these classes, the unweighted Maximum Stable Set (MS) Problem was known to be solvable in polynomial time by the struction method or by augmenting techniques, respectively, but the complexity of the MWS problem was open. A graph is nearly chordal if for each of its vertices, the subgraph induced by the set of its nonneighbors is chordal, and analogously for nearly perfect graphs.