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 -