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

Let G = (V , E) be a connected graph where V is the set of vertices of G and E is the set of edges of G. The distance from the vertex w ∈ V to the edge  e = u v ∈ E is given by d (e , w) = min { d (u , w) , d (v , w) }. A subset S e ⊂ V is called an edge metric generator for G if for every two distinct edges e 1 , e 2 ∈ E , there exists a vertex w ∈ S such that d (e 1 , w) ≠ d (e 2 , w). The edge metric generator with the minimum number of vertices is called an edge metric basis for G and the cardinality of the edge metric basis is called the edge metric dimension denoted by dim e (G). A subset S m ⊂ V is called a mixed metric generator for G if for every two distinct elements x , y ∈ V ∪ E , there exists a vertex w ∈ S such that d (x , w) ≠ d (y , w). A mixed metric generator containing a minimum number of vertices is called a mixed metric basis for G and the cardinality of a mixed metric basis is called the mixed metric dimension denoted by dim m (G). In this paper, we study the edge metric dimension and the mixed metric dimension of a plane graph  T n . [ABSTRACT FROM AUTHOR]