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

Abstract

Let $\Gamma=(V,E)$ be a connected graph. A vertex $i\in V$ recognizes two elements (vertices or edges) $j,k\in E \cap V$, if $d_{\Gamma}(i,j)\neq d_{\Gamma}(i,k)$. A set $S$ of vertices in a connected graph $\Gamma$ is a mixed metric generator for $\Gamma$ if every two distinct elements (vertices or edges)of $\Gamma$ are recognized by some vertex of $S$. The smallest cardinality of a mixed metric generator for $\Gamma$ is called the mixed metric dimension and is denoted by $\beta_{m}$. In this paper, the mixed metric dimension of generalized Petersen graph $P(n,2)$ is calculated. We established that generalized Petersen graph $P(n,2)$ has a mixed metric dimension equivalent to $4$ for $n\equiv 0,2(mod 4)$, and for $n\equiv 1,3(mod 4)$ the mixed metric dimension is $5$. Thus determining that each graph of the family of generalized Petersen graph $P(n,2)$ has a constant mixed metric dimension.