On Mixed Metric Dimension of Rotationally Symmetric Graphs

Abstract

A vertex u ∈ V(G) resolves (distinguish or recognize) two elements (vertices or edges) v, w ∈ E(G)UV(G) if dG(u, v) ≠ dG(u, w) . A subset Lm of vertices in a connected graph G is called a mixed metric generator for G if every two distinct elements (vertices and edges) of G are resolved by some vertex set of Lm. The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dimm(G). In this paper, we studied the mixed metric dimension for three families of graphs Dn, An, and Rn, known from the literature. We proved that, for Dn the dimm(Dn) = dime(Dn) = dim(Dn), when n is even, and for An the dimm(An) = dime(An), when n is even and odd. The graph Rn has mixed metric dimension 5.

Keywords

• Mixed metric dimension
• metric dimension
• edge metric dimension
• rotationally-symmetric