Connected Coloring Completion for General Graphs: Algorithms and Complexity

An r-component connected coloring of a graph is a coloring of the vertices so that each color class induces a subgraph having at most r connected components. The concept has been well-studied for r = 1, in the case of trees, under the rubric of convex coloring , used in modeling perfect phylogenies. Several applications in bioinformatics of connected coloring problems on general graphs are discussed, including analysis of protein-protein interaction networks and protein structure graphs, and of phylogenetic relationships modeled by splits trees. We investigate the r-COMPONENT CONNECTED COLORING COMPLETION (r-CCC) problem, that takes as input a partially colored graph, having k uncolored vertices, and asks whether the partial coloring can be completed to an r-component connected coloring. For r = 1 this problem is shown to be NPhard, but fixed-parameter tractable when parameterized by the number of uncolored vertices, solvable in time O∗(8k). We also show that the 1-CCC problem, parameterized (only) by the treewidth t of the graph, is fixed-parameter tractable; we show this by a method that is of independent interest. The r-CCC problem is shown to be W [1]-hard, when parameterized by the treewidth bound t, for any r ≥ 2. Our proof also shows that the problem is NP-complete for r = 2, for general graphs. Topics: Algorithms and Complexity, Bioinformatics ? This research has been supported by the Australian Research Council through the Australian Centre in Bioinformatics. The second and fourth authors also acknowledge the support provided by a William Best Fellowship at Grey College, Durham, while the paper was in preparation.