Connected coloring completion for general graphs: algorithms and complexity

  • Authors:

  • Benny Chor;Michael Fellows;Mark A. Ragan;Igor Razgon;Frances Rosamond;Sagi Snir

  • Affiliations:

  • Computer Science Department, Tel Aviv University, Tel Aviv, Israel;University of Newcastle, Callaghan NSW, Australia and Durham University, Institute of Advanced Study, Durham, United Kingdom;Institute for Molecular Biosciences, University of Queensland, Brisbane, QLD, Australia;Computer Science Department, University College Cork, Ireland;University of Newcastle, Callaghan NSW, Australia;Department of Mathematics, University of California, Berkeley

  • Venue:
  • COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
  • Year:
  • 2007

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Abstract

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.