On the computational complexity of Roman$\{2\}$-domination in grid graphs

Abstract

A Roman$\{2\}$-dominating function of a graph $G=(V,E)$ is a function $f:V\rightarrow \{0,1,2\}$ such that every vertex $x\in V$ with $f(x)=0$ either there exists at least one vertex $y\in N(x)$ with $f(y)=2$ or there are at least two vertices $u,v\in N(x)$ with $f(u)=f(v)=1$. The weight of a Roman$\{2\}$-dominating function f on G is defined to be the value of $\sum _{x\in V} f(x)$. The minimum weight of a Roman$\{2\}$-dominating function on G is called the Roman$\{2\}$-domination number of G. In this paper, we prove that the decision problem associated with Roman$\{2\}$-domination number is NP-complete even when restricted to subgraphs of grid graphs. Additionally, we answer an open question about the approximation hardness of Roman$\{2\}$-domination problem for bounded degree graphs.