Abstract
A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ge 2, we give an mathscr {O}^{star }(q^{{mathsf {tw}}})-time algorithm when the input graph is given together with one of its tree decompositions of width {mathsf {tw}} . We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is mathsf {XP} parameterized by clique-width.
Highlights
Vertex coloring problems are central in algorithmic graph theory, and appear in many variants
More precisely: we show that for any fixed q ≥ 2, qClique Coloring can be solved in time O, where tw denotes the width of a given tree decomposition of the input graph
We show that this running time is likely the best possible in this parameterization; we prove that under the Strong Exponential Time Hypothesis (SETH), for any q ≥ 2, there is no > 0 such that q- Clique Coloring can be solved in time O ((q − )tw)
Summary
Vertex coloring problems are central in algorithmic graph theory, and appear in many variants. For several graph classes it has been shown that all their members except odd cycles on at least five vertices (which require three colors) are 2-clique colorable [2,3,6,7,15,25,32,35] On these classes Clique Coloring is polynomial-time solvable. The doubleexponential dependence on w in the degree of the polynomial stems from the notorious property of clique colorings which we mentioned above; namely, that taking induced subgraphs does not necessarily preserve clique colorings. This results in a large amount of information that needs to be carried along as the algorithm progresses.
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