Edge dominating set and colorings on graphs with fixed clique-width Original Research Article

We consider both the vertex and the edge versions of three graph partitioning problems. These problems are dominating set, list-q-coloring with costs (fixed number of colors q) and chromatic number. 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 k, if the k-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). For the two list-q-coloring problems with costs, we even have linear algorithms. Since these classes of graphs include classes like P4-sparse graphs, distance hereditary graphs and graphs with bounded treewidth, our algorithms also apply to these graphs.