Resolving Roman domination in graphs
Let [Formula: see text] be a graph and [Formula: see text] be a Roman dominating function defined on [Formula: see text]. Let [Formula: see text] be some ordering of the vertices of [Formula: see text]. For any [Formula: see text], [Formula: see text] is defined by [Formula: see text]. If for all [Formula: see text], [Formula: see text], we have [Formula: see text], that is [Formula: see text], for some [Formula: see text], then [Formula: see text] is called a resolving Roman dominating function (RDF) on [Formula: see text]. The weight of a resolving RDF [Formula: see text] on [Formula: see text] is [Formula: see text]. The minimum weight of a resolving RDF on [Formula: see text] is called the resolving Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. A resolving RDF on [Formula: see text] with weight [Formula: see text] is called a [Formula: see text]-function on [Formula: see text]. In this paper, we find the resolving Roman domination number of certain well-known classes of graphs. We also categorize the class of graphs whose resolving Roman domination number equals their order.