On the complexity of minimum q-domination partization problems

Sayani Das () and Sounaka Mishra ()
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Sayani Das: IIT Madras
Sounaka Mishra: IIT Madras

Journal of Combinatorial Optimization, 2022, vol. 43, issue 2, No 4, 363-383

Abstract: 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 $$\chi _{dd}(G)$$ χ 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 $$\chi _{dd}(G[V{\setminus } S]) \le q$$ χ dd ( G [ V \ S ] ) ≤ q . For $$q=2$$ q = 2 , we prove that it is $${\mathsf {APX}}$$ APX -complete and is best approximable within a factor of 2. For $$q=3$$ q = 3 , it is approximable within a factor of $$O(\sqrt{\log n})$$ O ( log n ) and it is equivalent to minimum odd cycle transversal problem.

Keywords: Domination coloring; q-domination partization; Node deletion problem; Approximation algorithm (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10878-021-00779-1

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