AUTHOR=Raza Hassan , Ji Ying 

TITLE=Computing the Mixed Metric Dimension of a Generalized Petersen Graph P(n, 2)

JOURNAL=Frontiers in Physics

VOLUME=8

YEAR=2020

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.00211

DOI=10.3389/fphy.2020.00211

ISSN=2296-424X

ABSTRACT=<p>Let Γ = (<italic>V, E</italic>) be a connected graph. A vertex <italic>i</italic> ∈ <italic>V</italic> recognizes two elements (vertices or edges) <italic>j, k</italic> ∈ <italic>E</italic> ∩ <italic>V</italic>, if <italic>d</italic><sub>Γ</sub>(<italic>i, j</italic>) ≠ <italic>d</italic><sub>Γ</sub>(<italic>i, k</italic>). A set <italic>S</italic> of vertices in a connected graph Γ is a mixed metric generator for Γ if every two distinct elements (vertices or edges) of Γ are recognized by some vertex of <italic>S</italic>. The smallest cardinality of a mixed metric generator for Γ is called the mixed metric dimension and is denoted by β<sub><italic>m</italic></sub>. In this paper, the mixed metric dimension of a generalized Petersen graph <italic>P</italic>(<italic>n</italic>, 2) is calculated. We established that a generalized Petersen graph <italic>P</italic>(<italic>n</italic>, 2) has a mixed metric dimension equivalent to 4 for <italic>n</italic> ≡ 0, 2(<italic>mod</italic>4), and, for <italic>n</italic> ≡ 1, 3(<italic>mod</italic>4), the mixed metric dimension is 5. We thus determine that each graph of the family of a generalized Petersen graph <italic>P</italic>(<italic>n</italic>, 2) has a constant mixed metric dimension.</p><p><bold>2010 Mathematics Subject Classification:</bold> 05C12, 05C90</p>