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- W2571262586 abstract "A class domination coloring (also called as cd-coloring) of a graph is a proper coloring such that for every color class, there is a vertex that dominates it. The minimum number of colors required for a cd-coloring of the graph G, denoted by $$chi _{cd}(G)$$ , 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 $$mathcal {O}(2^n n^4 log n)$$ . Also, we show that the problem is $$mathsf {FPT}$$ with respect to the number of colors q as the parameter on chordal graphs. On graphs of girth at least 5, we show that the problem also admits a kernel with $$mathcal {O}(q^3)$$ vertices. For the second (deletion) problem, we show $$mathsf {NP}$$ -hardness for each $$q ge 2$$ . Further, on split graphs, we show that the problem is $$mathsf {NP}$$ -hard if q is a part of the input and $$mathsf {FPT}$$ with respect to k and q. As recognizing graphs with cd-chromatic number at most q is $$mathsf {NP}$$ -hard in general for $$q ge 4$$ , the deletion problem is unlikely to be $$mathsf {FPT}$$ when parameterized by the size of deletion set on general graphs. We show fixed parameter tractability for $$q in {2,3}$$ using the known algorithms for finding a vertex cover and an odd cycle transversal as subroutines." @default.
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- W2571262586 date "2017-01-01" @default.
- W2571262586 modified "2023-10-03" @default.
- W2571262586 title "Parameterized and Exact Algorithms for Class Domination Coloring" @default.
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- W2571262586 doi "https://doi.org/10.1007/978-3-319-51963-0_26" @default.
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