Edge metric dimension and mixed metric dimension of a plane graph Tn

Abstract

Let [Formula: see text] be a connected graph where [Formula: see text] is the set of vertices of [Formula: see text] and [Formula: see text] is the set of edges of [Formula: see text]. The distance from the vertex [Formula: see text] to the edge [Formula: see text] is given by [Formula: see text]. A subset [Formula: see text] is called an edge metric generator for [Formula: see text] if for every two distinct edges [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text]. The edge metric generator with the minimum number of vertices is called an edge metric basis for [Formula: see text] and the cardinality of the edge metric basis is called the edge metric dimension denoted by [Formula: see text]. A subset [Formula: see text] is called a mixed metric generator for [Formula: see text] if for every two distinct elements [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text]. A mixed metric generator containing a minimum number of vertices is called a mixed metric basis for [Formula: see text] and the cardinality of a mixed metric basis is called the mixed metric dimension denoted by [Formula: see text]. In this paper, we study the edge metric dimension and the mixed metric dimension of a plane graph [Formula: see text].