Parameterized and Exact Algorithms for Class Domination Coloring

A class domination coloring (also called 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 χcd(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. (1) Given a graph G, 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∗(2n) using polynomial method, where the O∗ notation suppresses polynomial factors. Also, we show that the problem is FPT when parameterized by the number of colors q on graphs of girth at least 5 and admits a kernel with O(q) vertices. For the second (deletion) problem, we show NP-hardness for each q ≥ 2. Further, on split graphs (for which cdchromatic number can be obtained in polynomial time), we show that it is NP-hard if q is a part of the input and FPT with respect to k and q. For q ≥ 4, as recognizing graphs with cd-chromatic number at most q is NP-hard on general graphs, the deletion problem is unlikely to be FPT when parameterized by the size of deletion set on general graphs. For the same parameterization, we show the fixed parameter tractability on general graphs for the case when q ∈ {2, 3}. Our algorithms use the known algorithms for finding a vertex cover and an odd cycle transversal as a subroutine.