On mixed metric dimension of some path related graphs

Abstract

A vertex k ∈ VG determined two elements (vertices or edges) `, m ∈ VG ∪ EG, if dG(k, `) 6= dG(k, m). A set Rm of vertices in a graph Gis a mixed metric generator for G, if two distinct elements (vertices or edges) are determined by some vertex set of Rm. The least number of elements in the vertex set of Rm is known as mixed metric dimension, and denoted as dimm(G). In this article, the mixed metric dimension of some path related graphs is obtained. Those path related graphs are P2n the square of a path, T(Pn) total graph of a path, the middle graph of a path M(Pn), and splitting graph of a path S(Pn). We proved that these families of graphs have constant and unbounded mixed metric dimension, respectively. We further presented an improved result for the metric dimension of the splitting graph of a path S(Pn).