Parameterized and exact algorithms for class domination coloring

A class domination coloring (also called cd-Coloring or dominated coloring) of a graph is proper in which every color contained the neighborhood some vertex. The minimum number colors required for any cd-coloring G, denoted by χcd(G), chromatic (cd-chromatic number) G. In this work, we consider two problems associated with context exact exponential-time algorithms and parameterized complexity. (1) Given G on n vertices, find its cd-chromatic number. (2) integers k q, can delete at most vertices such that resulting q? For first problem, give an algorithm running time O(2nn4logn). Also, show problem FPT respect to q as parameter chordal graphs. On graphs girth least 5, also admits kernel O(q3) vertices. second (deletion) NP-hardness each q≥2. Further, split graphs, NP-hard if part input combined parameters. As recognizing general q≥4, deletion unlikely be when size set We fixed tractability q∈{2,3} using known finding vertex cover odd cycle transversal subroutines.