On distance-s locating and distance-t dominating sets in graphs

Abstract

Let [Formula: see text] and [Formula: see text] be positive integers, and let [Formula: see text] be a simple and connected graph with vertex set [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the length of an [Formula: see text] geodesic in [Formula: see text] and let [Formula: see text]. Let [Formula: see text]. A set [Formula: see text] is a distance-[Formula: see text] locating set of [Formula: see text] if, for any distinct [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text], and the distance-[Formula: see text] location number, [Formula: see text], of [Formula: see text] is the minimum cardinality among all distance-[Formula: see text] locating sets of [Formula: see text]. A set [Formula: see text] is a distance-[Formula: see text] dominating set of [Formula: see text] if, for each vertex [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text], and the distance-[Formula: see text] domination number, [Formula: see text], of [Formula: see text] is the minimum cardinality among all distance-[Formula: see text] dominating sets of [Formula: see text]. The [Formula: see text]-location-domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality among all sets [Formula: see text] such that [Formula: see text] is both a distance-[Formula: see text] locating set and a distance-[Formula: see text] dominating set of [Formula: see text]. For any connected graph [Formula: see text] of order at least two, we show that [Formula: see text], where [Formula: see text]. We characterize connected graphs [Formula: see text] satisfying [Formula: see text] equals [Formula: see text] and [Formula: see text], respectively. We examine the relationship among [Formula: see text], [Formula: see text] and [Formula: see text]; along the way, we show that [Formula: see text] if [Formula: see text]. We also show that there exist graphs [Formula: see text] and [Formula: see text] with [Formula: see text] such that [Formula: see text] can be arbitrarily large. Moreover, we examine some graph classes.