On Mixed Metric Dimension of Rotationally Symmetric Graphs

A vertex u &#x2208; V(G) resolves (distinguish or recognize) two elements (vertices or edges) v, w &#x2208; E(G)UV(G) if d<sub>G</sub>(u, v) &#x2260; d<sub>G</sub>(u, w) . A subset L<sub>m</sub> of vertices in a connected graph G is called a mixed metr...

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Bibliographic Details
Main Authors: Hassan Raza, Jia-Bao Liu, Shaojian Qu
Format: Article
Language:English
Published: IEEE 2020-01-01
Series:IEEE Access
Subjects:
Online Access:https://ieeexplore.ieee.org/document/8937516/
Description
Summary:A vertex u &#x2208; V(G) resolves (distinguish or recognize) two elements (vertices or edges) v, w &#x2208; E(G)UV(G) if d<sub>G</sub>(u, v) &#x2260; d<sub>G</sub>(u, w) . A subset L<sub>m</sub> 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<sub>m</sub>. The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dim<sub>m</sub>(G). In this paper, we studied the mixed metric dimension for three families of graphs D<sub>n</sub>, A<sub>n</sub>, and R<sub>n</sub>, known from the literature. We proved that, for D<sub>n</sub> the dim<sub>m</sub>(D<sub>n</sub>) = dim<sub>e</sub>(D<sub>n</sub>) = dim(D<sub>n</sub>), when n is even, and for An the dim<sub>m</sub>(A<sub>n</sub>) = dim<sub>e</sub>(A<sub>n</sub>), when n is even and odd. The graph R<sub>n</sub> has mixed metric dimension 5.
ISSN:2169-3536