Generalized coloring for tree-like graphs Original Research Article

We discuss the Precoloring Extension (PrExt) and the List Coloring (LiCol) problems for trees, partial k-trees and cographs in the decision and the construction versions. Both problems for partial k-trees are solved in linear time when the number of colors is bounded by a constant and in polynomial time for an unbounded number of colors. For trees, we improve this to linear time. In contrast to that, the two problems differ in complexity for cographs. While PrExt has a linear-time decision algorithm, LiCol is shown to be NP-complete. We give polynomial-time algorithms for the corresponding enumeration problems # PrExt and # LiCol on partial k-trees and trees and for # PrExt on cographs.