On the metric dimension and fractional metric dimension for hierarchical product of graphs

A set of vertices $W$ {\em resolves} a graph $G$ if every vertex of $G$ is uniquely determined by its vector of distances to the vertices in $W$. The {\em metric dimension} for $G$, denoted by $\dim(G)$, is the minimum cardinality of a resolving set of $G$. In order to study the metric dimension for the hierarchical product $G_2^{u_2}\sqcap G_1^{u_1}$ of two rooted graphs $G_2^{u_2}$ and $G_1^{u_1}$, we first introduce a new parameter, the {\em rooted metric dimension} $\rdim(G_1^{u_1})$ for a rooted graph $G_1^{u_1}$. If $G_1$ is not a path with an end-vertex $u_1$, we show that $\dim(G_2^{u_2}\sqcap G_1^{u_1})=|V(G_2)|\cdot\rdim(G_1^{u_1})$, where $|V(G_2)|$ is the order of $G_2$. If $G_1$ is a path with an end-vertex $u_1$, we obtain some tight inequalities for $\dim(G_2^{u_2}\sqcap G_1^{u_1})$. Finally, we show that similar results hold for the fractional metric dimension.