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 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.
This research has been supported by the Australian Research Council through the Australian Centre in Bioinformatics. The second and fifth authors also acknowledge the support provided by a William Best Fellowship at Grey College, Durham, while the paper was in preparation.
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References
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991)
Alon, N., Yuster, R., Zwick, U.: Color-coding. Journal of the ACM 42, 844–856 (1995)
Bodlaender, H.L., Fellows, M., Langston, M., Ragan, M.A., Rosamond, F., Weyer, M.: Quadratic kernelization for convex recoloring of trees. In: Proceedings COCOON 2007, these proceedings (2007)
Bar-Yehuda, R., Feldman, I., Rawitz, D.: Improved approximation algorithm for convex recoloring of trees. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 55–68. Springer, Heidelberg (2006)
Borie, R.B., Parker, R.G., Tovey, C.A.: Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively generated graph families. Algorithmica 7, 555–581 (1992)
Bodlaender, H.L., Weyer, M.: Convex anc connected recolourings of trees and graphs. Manuscript (2005)
Bu, D., Zhao, Y., Cai, L., Xue, H., Zhu, X., Lu, H., Zhang, J., Sun, S., Ling, L., Zhang, N., Li, G., Chen, R.: Topological structure analysis of the protein-protein interaction network in budding yeast. Nucleic Acids Res. 31(9), 2443–2450 (2003)
Chen, J., Chor, B., Fellows, M., Huang, X., Juedes, D., Kanj, I., Xia, G.: Tight lower bounds for certain parameterized NP-hard problems. In: Proceedings of the 19th Annual IEEE Conference on Computational Complexity, pp. 150–160. IEEE Computer Society Press, Los Alamitos (2004)
Courcelle, B.: The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Fellows, M., Giannopoulos, P., Knauer, C., Paul, C., Rosamond, F., Whitesides, S., Yu, N.: The lawnmower and other problems: applications of MSO logic in geometry, Manuscript (2007)
Gramm, J., Nickelsen, A., Tantau, T.: Fixed-parameter algorithms in phylogenetics. Manuscript (2006)
Huson, D.H., Bryant, D.: Application of phylogenetic networks in evolutionary studies. Mol. Biol. E 23, 254–267 (2006)
Huson, D.H.: SplitsTree: a program for analyzing and visualizing evolutionary data. Bioinfomatics 14, 68–73 (1998)
Kelley, B.P., Sharan, R., Karp, R.M., Sittler, T., Root, D.E., Stockwell, B.R., Ideker, T.: Conserved pathways within bacteria and yeast as revealed by global protein network alignment. Proc. Natl. Acad. Sci. USA 100, 11394–11399 (2003)
Moran, S., Snir, S.: Convex recolorings of strings and trees: definitions, hardness results and algorithms. To appear in Journal of Computer and System Sciences. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 218–232. Springer, Heidelberg (2005) A preliminary version appeared
Moran, S., Snir, S., Sung, W.: Partial convex recolorings of trees and galled networks. Manuscript (2006)
Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford University Press, Oxford (2006)
Ramadan, E., Tarafdar, A., Pothen, A.: A hypergraph model for the yeast protein complex network. In: Fourth IEEE International Workshop on High Performance Computational Biology, Santa Fe, NM, April 26, 2004. IEEE Computer Society Press, Los Alamitos (2004)
Rual, J.F., Venkatesan, K., Hao, T., Hirozane-Kishikawa, T., Dricot, A., Li, N., Berriz, G.F., Gibbons, F.D., Dreze, M., Ayivi-Guedehoussou, N., Klitgord, N., Simon, C., Boxem, M., Milstein, S., Rosenberg, J., Goldberg, D.S., Zhang, L.V., Wong, S.L., Franklin, G., Li, S., Albala, J.S., Lim, J., Fraughton, C., Llamosas, E., Cevik, S., Bex, C., Lamesch, P., Sikorski, R.S., Vandenhaute, J., Zoghbi, H.Y., Smolyar, A., Bosak, S., Sequerra, R., Doucette-Stamm, L., Cusick, M.E., Hill, D.E., Roth, F.P., Vidal, M.: Nature 437, 1173–1178 (2005)
Schwikowski, B., Uetz, P., Fields, S.: A network of protein-protein interactions in yeast. Nature Biotechnology 18(12), 1257–1261 (2000)
Viveshwara, S., Brinda, K.V., Kannan, N.: Protein structure: insights from graph theory. J. Theoretical and Computational Chemistry 1, 187–211 (2002)
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Chor, B., Fellows, M., Ragan, M.A., Razgon, I., Rosamond, F., Snir, S. (2007). Connected Coloring Completion for General Graphs: Algorithms and Complexity. In: Lin, G. (eds) Computing and Combinatorics. COCOON 2007. Lecture Notes in Computer Science, vol 4598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73545-8_10
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DOI: https://doi.org/10.1007/978-3-540-73545-8_10
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