Dominator Coloring and CD Coloring in Almost Cluster Graphs
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In this paper, we study two popular variants of Graph Coloring – Dominator Coloring and CD Coloring. In both problems, we are given a graph G and a natural number ℓ as input and the goal is to properly color the vertices with at most ℓ colors with specific constraints. In Dominator Coloring, we require for each v ∈ V(G), a color c such that v dominates all vertices colored c. In CD Coloring, we require for each color c, a v ∈ V(G) which dominates all vertices colored c. These problems, defined due to their applications in social and genetic networks, have been studied extensively in the last 15 years. While it is known that both problems are fixed-parameter tractable (FPT) when parameterized by (t,ℓ) where t is the treewidth of G, we consider strictly structural parameterizations which naturally arise out of the problems' applications. We prove that Dominator Coloring is FPT when parameterized by the size of a graph's cluster vertex deletion (CVD) set and that CD Coloring is FPT parameterized by CVD set size plus the number of remaining cliques. En route, we design a simpler and faster FPT algorithms when the problems are parameterized by the size of a graph's twin cover, a special CVD set. When the parameter is the size of a graph's clique modulator, we design a randomized single-exponential time algorithm for the problems. These algorithms use an inclusion-exclusion based polynomial sieving technique and add to the growing number of applications using this powerful algebraic technique.
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