Algorithmic Aspects of Outer-Independent Double Roman Domination in Graphs

Let $G=(V,E)$ be graph. For any function $h:V\to \{0,1,2,3\}$, let $V_i=\{v\in V:h(v)=i\}$, $0\le i\le 3$. The function $h$ is called an outer-independent double Roman dominating function (OIDRDF) if the following conditions are satisfied.

(i)	If $v\in V_0$, then $|N(v)\cap V_3|\ge 1$ or $|N(v)\cap V_2|\ge 2$
(ii)	If $v\in V_1$, then $|N(v)\cap (V_2\cup V_3)|\ge 1$
(iii)	$V_0$ is independent.

The outer-independent double Roman domination number of $G$ is defined by ${\gamma }_{oidR}(G)=\mathrm{\text{min}}\left\{\right.{\sum }_{v\in V}h(v):h$ is an OIDRDF OF $G\left\}\right.$. We prove that the decision problem MOIDRDP, corresponding to ${\gamma }_{oidR}(G)$ is NP-complete for split graphs. We also show that it is linear time solvable for connected threshold graphs and bounded treewidth graphs. Finally, we show that the MOIDRDP and domination are not equivalent in computational complexity aspects.