algorithmic results on independent roman {2}-domination

An independent roman ${2}$-dominating function (ir2df) $f:v rightarrow {0, 1, 2}$ in a graph $g=(v,e)$ has the properties that $sum_{uin n(v)}f(u) geq 2$ if $f(v)=0$, and $f(u)=0$ for $u in n(v)$ if $f(v) geq 1$, where $v in v$. the weight of an ir2df in a graph $g$ is defined as the sum of its function values for all vertices, given by $omega(f)=sum_{vin v}f(v)$. the independent roman ${2}$-domination number of $g$, denoted $i_{{r2}}(g)$, is the minimum weight of all ir2dfs in $g$. in this paper, we prove that the independent roman ${2}$-domination problem (ir2d) is np-complete, even when restricted to chordal bipartite graphs. we then give an exact formula for the ir2d in corona graphs. finally, we present two linear-time algorithms for solving ir2d for proper interval graphs and block-cactus graphs, respectively.