Further results on (total) restrained Italian domination

Abstract

An Italian dominating function on a graph [Formula: see text] is defined as a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text] or at least two vertices [Formula: see text] for which [Formula: see text]. The weight of an Italian dominating function is [Formula: see text]. The Italian domination number is the minimum weight taken over all Italian dominating functions of [Formula: see text] and denoted by [Formula: see text]. Three domination parameters related to the Italian dominating function are total Italian, restrained Italian, and total restrained Italian dominating function. A total ((restrained) (total restrained)) Italian dominating function [Formula: see text] is an Italian dominating function if the set of vertices with positive label ((the set of vertices with label [Formula: see text]), (at the same time, the set of vertices with positive label and the set of vertices with label [Formula: see text])) induces ((induces) (induce)) a subgraph with no isolated vertex. A minimum weight of any total ((restrained) (total restrained)) Italian dominating function [Formula: see text] is called a total ((restrained) (total restrained)) Italian domination number denoted by [Formula: see text], (([Formula: see text]) ([Formula: see text])). We initiate the study of parameters, restrained and total restrained Italian domination number of a graph [Formula: see text] and the middle graph of [Formula: see text]. For the family of standard graphs, we obtain the exact value of these parameters. For arbitrary graph [Formula: see text], we obtain the sharp bounds of these parameters, and for some corona graphs, we establish the precise value of these parameters.