Parameterized and exact algorithms for class domination coloring.

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 neighborhood of some vertex. The minimum number of colors required for any cd-coloring of 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 O(2(n)n(4) log n). Also, we show that the problem is FPT 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 O(q(3)) vertices. For the second (deletion) problem, we show NP-hardness for each q >= 2. Further, on split graphs, we show that the problem is NP-hard if q is a part of the input and FPT with respect to k and q as combined parameters. As recognizing graphs with cd-chromatic number at most q is NP-hard in general for q >= 4, the deletion problem is unlikely to be FPT when parameterized by the size of the deletion set on general graphs. We show fixed parameter tractability for q is an element of {2, 3} using the known algorithms for finding a vertex cover and an odd cycle transversal as subroutines. (C) 2020 Elsevier B.V. All rights reserved.