Restrained Italian domination in graphs

TL;DR

This paper introduces the concept of restrained Italian domination in graphs, studies its properties, proves NP-hardness of computing it, and characterizes extremal trees and bounds for various graphs.

Contribution

It defines the restrained Italian domination number, proves its NP-hardness, and characterizes extremal trees and bounds for this new parameter.

Findings

NP-hardness of computing the parameter even for special graph classes

Lower bounds on the parameter for trees, with characterization of extremal cases

Sharp bounds and characterizations for general graphs

Abstract

For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) $f:V(G)\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V(G)$ with $f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$ or $v$ is adjacent to least two vertices assigned $1$ under $f$. The weight of an ID function is $\sum_{v\in V(G)}f(v)$. The Italian domination number is the minimum weight taken over all ID functions of $G$. In this paper, we initiate the study of a variant of ID functions. A restrained Italian dominating function (RID function) $f$ of $G$ is an ID function of $G$ for which the subgraph induced by $\{v\in V(G)\mid f(v)=0\}$ has no isolated vertices, and the restrained Italian domination number $\gamma_{rI}(G)$ is the minimum weight taken over all RID functions of $G$. We first prove that the problem of computing this parameter is NP-hard, even when restricted to bipartite graphs and chordal graphs as well as planar graphs with maximum degree five. We prove that $\gamma_{rI}(T)$ for a tree $T$ of order $n\geq3$ different from the double star $S_{2,2}$ can be bounded from below by $(n+3)/2$. Moreover, all extremal trees for this lower bound are characterized in this paper. We also give some sharp bounds on this parameter for general graphs and give the characterizations of graphs $G$ with small or large $\gamma_{rI}(G)$.