Parameterized and Exact Algorithms for Class Domination Coloring
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A class domination coloring (also called cd-Coloring or dominated coloring) of a graph is a proper coloring in which every color class is contained in the neighbourhood of some vertex. The minimum number of colors required for any cd-coloring of G, denoted by χ_cd(G), is called the class domination chromatic number (cd-chromatic number) of G. In this work, we consider two problems associated with the cd-coloring of a graph in the context of exact exponential-time algorithms and parameterized complexity. (1) Given a graph G on n vertices, find its cd-chromatic number. (2) Given a graph G and integers k and q, can we delete at most k vertices such that the cd-chromatic number of the resulting graph is at most q? For the first problem, we give an exact algorithm with running time (2^n n^4 log n). Also, we show that the problem is with respect to the number q of colors as the parameter on chordal graphs. On graphs of girth at least 5, we show that the problem also admits a kernel with (q^3) vertices. For the second (deletion) problem, we show -hardness for each q ≥ 2. Further, on split graphs, we show that the problem is -hard if q is a part of the input and with respect to k and q as combined parameters. As recognizing graphs with cd-chromatic number at most q is -hard in general for q ≥ 4, the deletion problem is unlikely to be when parameterized by the size of the deletion set on general graphs. We show fixed parameter tractability for q ∈{2,3} using the known algorithms for finding a vertex cover and an odd cycle transversal as subroutines.
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