On the Local Metric Dimension of Line Graphs

Abstract

Let [Formula: see text] be a graph. For any [Formula: see text], if there exists [Formula: see text] such that [Formula: see text], we say that [Formula: see text] resolving [Formula: see text]. A set [Formula: see text] of vertices in [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 study the relation between [Formula: see text] and [Formula: see text]. Furthermore, we construct a graph [Formula: see text] such that [Formula: see text] and [Formula: see text]. Finally, we investigate the local metric dimension of several special line graphs.