Abstract
In this paper we initiate a systematic study of a problem that has the flavor of two classical problems, namely Coloring and Domination, from the perspective of algorithms and complexity. A dominator coloring of a graph G is an assignment of colors to the vertices of G such that it is a proper coloring and every vertex dominates all the vertices of at least one color class. The minimum number of colors required for a dominator coloring of G is called the dominator chromatic number of G and is denoted by χ d (G). In the Dominator Coloring (DC) problem, a graph G and a positive integer k are given as input and the objective is to check whether χ d (G) ≤ k. We first show that unless P=NP, DC cannot be solved in polynomial time on bipartite, planar, or split graphs. This resolves an open problem posed by Chellali and Maffray [Dominator Colorings in Some Classes of Graphs, Graphs and Combinatorics, 2011] about the polynomial time solvability of DC on chordal graphs. We then complement these hardness results by showing that the problem is fixed parameter tractable (FPT) on chordal graphs and in graphs which exclude a fixed apex graph as a minor.
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References
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. Journal of Algorithms 12(2), 308–340 (1991)
Arumugam, S., Bagga, J., Chandrasekar, K.R.: On dominator colorings in graphs (2010) (manuscript)
Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)
Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel Bounds for Disjoint Cycles and Disjoint Paths. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 635–646. Springer, Heidelberg (2009)
Borie, R.B., Parker, G.R., Tovey, C.A.: Automatic Generation of Linear-Time Algorithms from Predicate Calculus Descriptions of Problems on Recursively Constructed Graph Families. Algorithmica 7, 555–581 (1992)
Chellali, M., Maffray, F.: Dominator colorings in some classes of graphs. Graphs and Combinatorics, 1–11 (2011)
Courcelle, B.: The monadic second-order logic of graphs. i. recognizable sets of finite graphs. Information and Computation 85(1), 12–75 (1990)
Courcelle, B.: The expression of graph properties and graph transformations in monadic second-order logic. In: Rozenberg, G. (ed.) Handbook of Graph Grammars and Computing by Graph Transformations: Foundations, ch. 5, vol. 1. World Scientific (1997)
Courcelle, B., Mosbah, M.: Monadic second-order evaluations on tree-decomposable graphs. Theoretical Computer Science 109(1-2), 49–82 (1993)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)
Downey, R.G., Fellows, M.R., McCartin, C., Rosamond, F.: Parameterized approximation of dominating set problems. Information Processing Letters 109(1), 68–70 (2008)
Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45(4), 634–652 (1998)
Feige, U., Hajiaghayi, M., Lee, J.R.: Improved approximation algorithms for minimum-weight vertex separators. SIAM Journal on Computing 38(2), 629–657 (2008)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)
Fomin, F.V., Golovach, P., Thilikos, D.M.: Contraction Bidimensionality: The Accurate Picture. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 706–717. Springer, Heidelberg (2009)
Fomin, F.V., Thilikos, D.M.: Dominating sets in planar graphs: Branch-width and exponential speed-up. SIAM Journal on Computing 36(2), 281–309 (2006)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP–Completeness. Freeman, San Francisco (1979)
Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoretical Computer Science 1, 237–267 (1976)
Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM Journal on Computing 1(2), 180–187 (1972)
Gera, R.: On dominator coloring in graphs. In: Graph Theory Notes of New York, pp. 25–30. LII (2007)
Gera, R.: On the dominator colorings in bipartite graphs. In: ITNG, pp. 1–6. IEEE (2007)
Gera, R., Rasmussen, C., Horton, S.: Dominator colorings and safe clique partitions. Congressus Numerantium 181(7-9), 19–32 (2006)
Halldórsson, M.M.: A still better performance guarantee for approximate graph coloring. Information Processing Letters 45(1), 19–23 (1993)
Hedetniemi, S., Hedetniemi, S., McRae, A., Blair, J.: Dominator colorings of graphs (2006) (preprint)
Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9(3), 256–278 (1974)
Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Mathematics 13, 383–390 (1975)
Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. Journal of the ACM 41(5), 960–981 (1994)
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Arumugam, S., Chandrasekar, K.R., Misra, N., Philip, G., Saurabh, S. (2011). Algorithmic Aspects of Dominator Colorings in Graphs. In: Iliopoulos, C.S., Smyth, W.F. (eds) Combinatorial Algorithms. IWOCA 2011. Lecture Notes in Computer Science, vol 7056. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25011-8_2
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