Restrained double Italian domination in graphs

Abstract

Let $G$ be a graph with vertex set $V(G)$.A double Italian dominating function (DIDF) is a function $f:V(G)longrightarrow {0,1,2,3}$having the property that $f(N[u])geq 3$ for every vertex $uin V(G)$ with $f(u)in {0,1}$,where $N[u]$ is the closed neighborhood of $u$. If $f$ is a DIDF on $G$, then let $V_0={vin V(G): f(v)=0}$. A restrained double Italian dominating function (RDIDF)is a double Italian dominating function $f$ having the property that the subgraph induced by $V_0$ does not have an isolated vertex.The weight of an RDIDF $f$ is the sum $sum_{vin V(G)}f(v)$, and the minimum weight of an RDIDF on a graph $G$ is the restrained double Italian domination number.We present bounds and Nordhaus-Gaddum type results for the restrained double Italian domination number. In addition, we determine therestrained double Italian domination number for some families of graphs.