The monadic second-order logic of graphs. I. recognizable sets of finite graphs
Information and Computation
Easy problems for tree-decomposable graphs
Journal of Algorithms
Monadic second-order evaluations on tree-decomposable graphs
Theoretical Computer Science - Special issue on selected papers of the International Workshop on Computing by Graph Transformation, Bordeaux, France, March 21–23, 1991
A still better performance guarantee for approximate graph coloring
Information Processing Letters
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
The expression of graph properties and graph transformations in monadic second-order logic
Handbook of graph grammars and computing by graph transformation
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
On the hardness of approximating minimization problems
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Dominating Sets in Planar Graphs: Branch-Width and Exponential Speed-Up
SIAM Journal on Computing
On the Dominator Colorings in Bipartite Graphs
ITNG '07 Proceedings of the International Conference on Information Technology
Improved Approximation Algorithms for Minimum Weight Vertex Separators
SIAM Journal on Computing
Parameterized approximation of dominating set problems
Information Processing Letters
Approximation algorithms for combinatorial problems
Journal of Computer and System Sciences
Dominator Colorings in Some Classes of Graphs
Graphs and Combinatorics
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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 χd(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 χd(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.