Parameterized and Exact Algorithms for Class Domination Coloring

R. Krithika, Ashutosh Rai, Saket Saurabh, Prafullkumar Tale

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.