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, Maximum Clique, Graph Coloring and Minimum Fill-in, but few examples of graph classes having clique separators are known. We use this method to solve MWS in polynomial time for two classes where the unweighted Maximum Stable Set (MS) Problem is solvable in polynomial time by augmenting techniques but the complexity of the MWS problem was open. Another example, namely a result by Alekseev for the MWS problem on a subclass of P5-free graphs obtained by clique separators, can be improved by our techniques. We also combine clique separators with decomposition by homogeneous sets in graphs and use the following notion: A graph is nearly Π if for each of its vertices, the subgraph induced by the set of its nonneighbors has property Π. We deal with the cases . This also simplifies a result obtained by a method called struction.