On mixed metric dimension in subdivision, middle, and total graphs

Abstract

Let $G$ be a graph and let $S(G)$, $M(G)$, and $T(G)$ be the subdivision, the middle, and the total graph of $G$, respectively. Let ${\rm dim}(G)$, ${\rm edim}(G)$, and ${\rm mdim}(G)$ be the metric dimension, the edge metric dimension, and the mixed metric dimension of $G$, respectively. In this paper, for the subdivision graph it is proved that $\frac{1}{2}\max\{{\rm dim}(G),{\rm edim}(G)\}\leq{\rm mdim}(S(G))\leq{\rm mdim}(G)$. A family of graphs $G_n$ is constructed for which ${\rm mdim}(G_n)-{\rm mdim}(S(G_n))\ge 2$ holds and this shows that the inequality ${\rm mdim}(S(G))\leq{\rm mdim}(G)$ can be strict, while for a cactus graph $G$, ${\rm mdim}(S(G))={\rm mdim}(G)$. For the middle graph it is proved that ${\rm dim}(M(G))\leq{\rm mdim}(G)$ holds, and if $G$ is tree with $n_1(G)$ leaves, then ${\rm dim}(M(G))={\rm mdim}(G)=n_1(G)$. Moreover, for the total graph it is proved that ${\rm mdim}(T(G))=2n_1(G)$ and ${\rm dim}(G)\leq{\rm dim}(T(G))\leq n_1(G)$ hold when $G$ is a tree.