Dominator coloring and CD coloring in almost cluster graphs

In this paper, we study two variants of Coloring - Dominator Coloring and Class Domination Coloring. In both problems, we are given a graph G and a $\ell \in N$ and the goal is to properly color the vertices with at most ℓ colors. In Dominator Coloring, we require for each $v\in V\left(G\right)$, a color c such that v dominates all vertices colored c. In Class Domination Coloring, we require for each color c, a $v\in V\left(G\right)$ which dominates all vertices colored c. We prove that Dominator Coloring is FPT when parameterized by the size of a graph's CVD set and that Class Domination Coloring is FPT parameterized by CVD set size plus the number of remaining cliques. En route, we design simpler algorithms when the problems are parameterized by the size of a graph's twin cover. When the parameter is the size of a graph's clique modulator, we design a randomized single-exponential time algorithm.