Hardness Results of Global Roman Domination in Graphs

Abstract

A Roman dominating function (RDF) of a graph $$ G=(V,E) $$ is a function $$f : V \rightarrow \{0, 1, 2\}$$ such that every vertex assigned the value 0 is adjacent to a vertex assigned the value 2. A global Roman dominating function (GRDF) of a graph $$ G=(V,E) $$ is a function $$f : V \rightarrow \{0, 1, 2\}$$ such that f is a Roman dominating function of both G and its complement $$ \overline{G} $$ . The weight of f is $$f(V) =\varSigma _{u\in V} f(u)$$ . The minimum weight of a GRDF in a graph G is known as global Roman domination number of G and is denoted by $$\gamma _{gR}(G)$$ . Minimum Global Roman Domination is to find a global Roman dominating function of minimum weight and Decide Global Roman Domination is the decision version of Minimum Global Roman Domination. In this paper, we show that Decide Global Roman Domination is NP-complete for bipartite graphs and chordal graphs. We also show that Minimum Global Roman Domination cannot be approximated within a factor of $$ (\frac{1}{2}-\epsilon )\ln \vert V \vert $$ for any $$ \epsilon >0 $$ unless $$ \textsf {P}=\textsf {NP} $$ . On the positive side, we propose an $$O(\ln |V|)$$ -approximation algorithm for Minimum Global Roman Domination for any graph $$G=(V,E)$$ .