Parameterized and exact algorithms for class domination coloring

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 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 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(2nn4 log n). Also, we show that the problem is 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 O(q3) 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 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 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.