A Note on Roman \{2\}-domination problem in graphs

Abstract: For a graph $G=(V,E)$, a Roman ${2}$-dominating function (R2DF)$f:V\rightarrow {0,1,2}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either there exists a neighbor $u\in N(v)$, with $f(u)=2$, or at least two neighbors $x,y\in N(v)$ having $f(x)=f(y)=1$. The weight of a R2DF is the sum $f(V)=\sum_{v\in V}{f(v)}$, and the minimum weight of a R2DF is the Roman ${2}$-domination number $\gamma_{{R2}}(G)$. A R2DF is independent if the set of vertices having positive function values is an independent set. The independent Roman ${2}$-domination number $i_{{R2}}(G)$ is the minimum weight of an independent Roman ${2}$-dominating function on $G$. In this paper, we show that the decision problem associated with $\gamma_{{R2}}(G)$ is NP-complete even when restricted to split graphs. We design a linear time algorithm for computing the value of $i_{{R2}}(T)$ for any tree $T$. This answers an open problem raised by Rahmouni and Chellali [Independent Roman ${2}$-domination in graphs, Discrete Applied Mathematics 236 (2018), 408-414]. Chellali, Haynes, Hedetniemi and McRae \cite{chellali2016roman} have showed that Roman ${2}$-domination number can be computed for the class of trees in linear time. As a generalization, we present a linear time algorithm for solving the Roman ${2}$-domination problem in block graphs.