A characterization of some graphs with metric dimension two

Abstract

A set [Formula: see text] is called a resolving set, if for each pair of distinct vertices [Formula: see text] there exists [Formula: see text] such that [Formula: see text], where [Formula: see text] is the distance between vertices [Formula: see text] and [Formula: see text]. The cardinality of a minimum resolving set for [Formula: see text] is called the metric dimension of [Formula: see text] and is denoted by [Formula: see text]. A [Formula: see text]-tree is a chordal graph all of whose maximal cliques are the same size [Formula: see text] and all of whose minimal clique separators are also all the same size [Formula: see text]. A [Formula: see text]-path is a [Formula: see text]-tree with maximum degree [Formula: see text], where for each integer [Formula: see text], [Formula: see text], there exists a unique pair of vertices, [Formula: see text] and [Formula: see text], such that [Formula: see text]. In this paper, we prove that if [Formula: see text] is a [Formula: see text]-path, then [Formula: see text]. Moreover, we provide a characterization of all [Formula: see text]-trees with metric dimension two.