The Local Metric Dimension and Distance-Edge-Monitoring Number of Graph

Abstract

Let [Formula: see text] be a graph and [Formula: see text]. The vertex [Formula: see text] is said to resolve a pair [Formula: see text] and [Formula: see text] if and only if [Formula: see text]. A set [Formula: see text] is defined as a resolving set of [Formula: see text] if for all [Formula: see text], the pair [Formula: see text] is resolved by some [Formula: see text]. The minimum cardinality of a resolve set of [Formula: see text] is defined as [Formula: see text]. A set [Formula: see text] is a local resolve set of [Formula: see text] if for all [Formula: see text] such that [Formula: see text], the pair [Formula: see text] is resolved by some [Formula: see text]. The minimum cardinality of a local resolve set of [Formula: see text] is defined as [Formula: see text]. An edge [Formula: see text] is said to be monitored by [Formula: see text] if [Formula: see text] or [Formula: see text]. A set [Formula: see text] is a distance-edge-monitoring (DEM) set if for all [Formula: see text], [Formula: see text] is monitored by some [Formula: see text]. The minimum cardinality of a [Formula: see text] set of [Formula: see text] is defined as [Formula: see text]. In this paper, we obtained that [Formula: see text] for all non trivial graphs with order [Formula: see text], and the exact value of [Formula: see text] and [Formula: see text] for [Formula: see text], [Formula: see text]. Also, we obtained that if [Formula: see text] then [Formula: see text]. With respect to the relation between the defined graph invariants, it was proved a bound for [Formula: see text] for [Formula: see text] and the exact values of [Formula: see text] for [Formula: see text], where [Formula: see text] (resp, [Formula: see text])is the maximum value of [Formula: see text] (resp, [Formula: see text]) over all graphs [Formula: see text] with order [Formula: see text]. Finally, we proved that for [Formula: see text], there exists a graph with order [Formula: see text] such that [Formula: see text] and [Formula: see text].