On the (total) restrained double Italian domination of central of graphs

Abstract

Let [Formula: see text] be a graph with [Formula: see text] and [Formula: see text]. A double Italian dominating function (DIDF) of [Formula: see text] is a function [Formula: see text] having the property that for every vertex [Formula: see text], [Formula: see text], if [Formula: see text], and [Formula: see text], if [Formula: see text]. The weight of a DIDF is [Formula: see text]. The double Italian domination number is the minimum weight taken over all DIDFs of [Formula: see text] and denoted by [Formula: see text]. Two domination parameters related to the DIDF are restrained, and total restrained DID function which are studied on some graphs. A central graph [Formula: see text] of a graph [Formula: see text] is obtained by subdividing each edge of [Formula: see text] exactly once and joining all the non-adjacent vertices of [Formula: see text]. In this paper, we obtain the exact value for restrained and total restrained of double Italian domination number of central of custom graphs and in addition, we determine the sharp bounds for central of any graph.