Abstract
A dominator coloring of a graph [Formula: see text] is a proper vertex coloring with a domination property that each vertex dominates a color class. The minimum number of colors used in a dominator coloring of [Formula: see text] is called dominator chromatic number of [Formula: see text] and is denoted as [Formula: see text]. Graphs with [Formula: see text] have characterizations and such graphs can be recognized in polynomial time. However, for [Formula: see text], deciding whether [Formula: see text] is NP-hard. In this paper, we investigate the computational complexity of minimum [Formula: see text]-dominator partization problem (Min-[Formula: see text]-D-Partz). In Min-[Formula: see text]-D-Partz, given a graph [Formula: see text], the objective is to find a vertex set [Formula: see text] of minimum size such that [Formula: see text]. We prove that Min-[Formula: see text]-D-Partz is APX-complete and has a two-factor approximation algorithm. We also prove that Min-[Formula: see text]-D-Partz cannot be approximated within any constant factor and can be approximated within a factor of [Formula: see text].
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