On Mixed Metric Dimension of Rotationally Symmetric Graphs

Abstract

A vertex u\in V(G) resolves (distinguish or recognize) two elements (vertices or edges) v,w\in E(G) \cup V(G) if d-{G}(u,v)eq d-{G}(u,w). A subset L- {\textrm {m}} 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 L- {\textrm {m}}. The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dim-{m}(G). In this paper, we studied the mixed metric dimension for three families of graphs {D}-{n} , {A}-{n} , and {R}-{n} , known from the literature. We proved that, for {D}-{n} the dim-{m}({D}-{n})=dim-{e}({D}-{n})=dim({D}-{n}) , when n is even, and for {A}-{n} the dim-{m}({A}-{n})=dim-{e}({A}-{n}) , when n is even and odd. The graph {R}-{n} has mixed metric dimension 5.