Connected coloring completion for general graphs: Algorithms and complexity

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

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 NP-hard, 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.

Original languageEnglish
Title of host publicationComputing and Combinatorics - 13th Annual International Conference, COCOON 2007, Proceedings
PublisherSpringer Verlag
Number of pages11
ISBN (Print)9783540735441
StatePublished - 2007
Externally publishedYes
Event13th Annual International Computing and Combinatorics Conference, COCOON 2007 - Banff, Canada
Duration: 16 Jul 200719 Jul 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4598 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference13th Annual International Computing and Combinatorics Conference, COCOON 2007

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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