Edge metric dimension and mixed metric dimension of planar graph Qn

Abstract

Let G be a finite, simple, and connected graph with the vertex set V and the edge set E . The distance between the vertex u and the edge e = v w is defined as d ( u , e ) = min { d ( u , v ) , d ( u , w ) } . A vertex x distinguishes two edges e 1 , e 2 if d ( x , e 1 ) ≠ d ( x , e 2 ) . A subset L e of V is called an edge metric generator for G if every two distinct edges of G are distinguished by some vertex of L e . The minimum cardinality of an edge metric generator for G is called the edge metric dimension and is denoted by d i m e ( G ) . Similarly, a vertex x distinguishes two elements (vertices or edges) u , v ∈ V ∪ E if d ( x , u ) ≠ d ( x , v ) . A subset L m of V is called a mixed metric generator for G if every two distinct elements (vertices and edges) of G are distinguished by some vertex of L m . The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by d i m m ( G ) . In this paper, we study the edge metric dimension and mixed metric dimension of planar graph Q n . We prove that the edge metric dimension and the mixed metric dimension of Q n are both finite and do not depend upon the number of vertices.