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