# Restrained double Italian domination in graphs

Document Type : Original paper

Author

RWTH Aachen University

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 $u\in V(G)$ with $f(u)\in \{0,1\}$, where $N[u]$ is the closed neighborhood of $u$. If $f$ is a DIDF on $G$, then let $V_0=\{v\in V(G): f(v)=0\}$. 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 $\sum_{v\in V(G)}f(v)$, 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 the restrained double Italian domination number for some families of graphs.

Keywords

Main Subjects

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