On the Local Metric Dimension of Graphs

Abstract

Let [Formula: see text] be a graph. A set [Formula: see text] is a local resolving set of [Formula: see text] if there exists [Formula: see text] such that [Formula: see text] for any [Formula: see text]. The local metric dimension [Formula: see text] of [Formula: see text] is the minimum cardinality of all the local resolving sets of [Formula: see text]. In this paper, we characterize the graphs with [Formula: see text]. Next, we obtain the Nordhaus–Gaddum-type results for local metric dimension. Finally, the local metric dimension of several graph classes is given.