On the complexity of minimum q-domination partization problems

Abstract

A domination coloring of a graph G is a proper vertex coloring with an additional condition that each vertex dominates a color class and each color class is dominated by a vertex. The minimum number of colors used in a domination coloring of G is denoted as $\chi _{dd}(G)$ and it is called domination chromatic number of G. In this paper, we give a polynomial-time characterization of graphs with domination chromatic number at most 3 and consider the approximability of a node deletion problem called minimum q-domination partization. Given a graph G, in the Minimum q-Domination Partization problem (in short Min-q-Domination-Partization), the objective is to find a vertex set S of minimum size such that $\chi _{dd}(G[V{\setminus } S]) \le q$. For $q=2$, we prove that it is ${\mathsf {APX}}$-complete and is best approximable within a factor of 2. For $q=3$, it is approximable within a factor of $O(\sqrt{\log n})$ and it is equivalent to minimum odd cycle transversal problem.