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In this paper, we consider a large class of vertex partitioning problems and apply to those the theory of algorithm design for problems restricted to partial k-trees. We carefully describe the details of algorithms and analyze their complexity in an attempt to make the algorithms feasible as solutions for practical applications. We give a precise characterization of vertex partitioning problems, which include domination, coloring and packing problems and their variants. Several new graph parameters are introduced as generalizations of classical parameters. This characterization provides a basis for a taxonomy of a large class of problems, facilitating their common algorithmic treatment and allowing their uniform complexity classi cation. We present a design methodology of practical solution algorithms for generally NP-hard problems when restricted to partial k-trees (graphs with treewidth bounded by k). This \practicality" accounts for dependency on the parameter k of the computational complexity of the resulting algorithms. By adapting the algorithm design methodology on partial k-trees to vertex partitioning problems, we obtain the rst algorithms for these problems with reasonable time complexity as a function of treewidth. As an application of the methodology, we give the rst polynomial-time algorithm on partial k-trees for computation of the Grundy Number.

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