The dominant edge metric dimension of graphs

Abstract

For an ordered subset S = { v 1 , …, v k } of vertices in a connected graph G and an edge e ′ of G , the edge metric S -representation of e ′= a b is the vector r G e ( e ′| S )=( d G ( e ′, v 1 ),…, d G ( e ′, v k )) , where d G ( e ′, v i )=min{ d G ( a , v i ), d G ( b , v i )} . A dominant edge metric generator for G is a vertex cover S of G such that the edges of G have pairwise different edge metric S -representations. A dominant edge metric generator of smallest size of G is called a dominant edge metric basis for G . The size of a dominant edge metric basis of G is denoted by D d i m e ( G ) and is called the dominant edge metric dimension. In this paper, the concept of dominant edge metric dimension (DEMD for short) is introduced and its basic properties are studied. Moreover, NP-hardness of computing DEMD of connected graphs is proved. Furthermore, this invariant is investigated under some graph operations at the end of the paper.