Polynomial algorithms for partitioning problems on graphs with fixed clique-width (Extended abstract)

We consider three graph partitioning problems, both from the vertices and the edges point of view. These problems are dominating set, list-q-coloring with costs (fixed number of colors q) and coloring with non-fixed number of colors. They are all known to be NP-hard in general. We show that all these problems (except edge-coloring) can be solved in polynomial time on graphs with clique-width bounded by some constant kappa, if the kappa-expression of the input graph is also given. In particular, we present the first polynomial algorithms (on these classes) for chromatic number, edge-dominating set and list-q-coloring with costs (fixed number of colors q, both vertex and edge versions). Since these classes of graphs include classes like P-4-sparse graphs, distance hereditary graphs and graphs with bounded treewidth, our algorithms also apply to these graphs.