Outer independent double Roman domination number of graphs

A double Roman dominating function of a graph $G$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that for each vertex $v$ with $f(v)=0$, there exists $u\in N(v)$ with $f(u)=3$, or there are $u,w\in N(v)$ with $f(u)=f(w)=2$, and if $f(v)=1$, then $v$ is adjacent to a vertex assigned at least $2$ under $f$. The double Roman domination number $\gamma_{dR}(G)$ is the minimum weight $f(V(G))=\sum_{v\in V(G)}f(v)$ among all double Roman dominating functions of $G$. An outer independent double Roman dominating function is a double Roman dominating function $f$ for which the set of vertices assigned $0$ under $f$ is independent. The outer independent double Roman domination number $\gamma_{oidR}(G)$ is the minimum weight taken over all outer independent double Roman dominating functions of $G$. In this work, we present some contributions to the study of outer independent double Roman domination in graphs. Characterizations of the families of all connected graphs with small outer independent double Roman domination numbers, and tight lower and upper bounds on this parameter are given. We moreover bound this parameter for a tree $T$ from below by two times the vertex cover number of $T$ plus one. We also prove that the decision problem associated with $\gamma_{oidR}(G)$ is NP-complete even when restricted to planar graphs with maximum degree at most four. Finally, we give an exact formula for this parameter concerning the corona graphs.