TY - GEN

T1 - Connected coloring completion for general graphs

T2 - 13th Annual International Computing and Combinatorics Conference, COCOON 2007

AU - Chor, Benny

AU - Fellows, Michael

AU - Ragan, Mark A.

AU - Razgon, Igor

AU - Rosamond, Frances

AU - Snir, Sagi

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=37849005778&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-73545-8_10

DO - 10.1007/978-3-540-73545-8_10

M3 - Conference contribution

AN - SCOPUS:37849005778

SN - 9783540735441

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 75

EP - 85

BT - Computing and Combinatorics - 13th Annual International Conference, COCOON 2007, Proceedings

PB - Springer Verlag

Y2 - 16 July 2007 through 19 July 2007

ER -