On the Algorithmic Complexity of Roman $\{2\}$-Domination (Italian Domination)

Abstract

A Roman $\{2\}$-dominating function (R2DF) $f:V\longrightarrow \{0,1,2\}$ of a graph $G=(V, E)$ has the property that for every vertex $v\in V$ with $f(v)=0$ either there is a vertex $u\in N(v)$ with $f(u)=2$ or there are two vertices $x,y\in N(v)$ with $f(x)=f(y)=1$. The weight of f is the sum $f(V)=\sum _{v\in V}f (v)$. The minimum weight of an R2DF on G is the Roman $\{2\}$-domination number of G. In this paper, we first show that the associated decision problem for Roman $\{2\}$-domination is NP-complete even when restricted to planar graphs. Then, we give a linear algorithm that computes the Roman $\{2\}$-domination number of a given unicyclic graph.