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

TL;DR

This paper investigates the metric and fractional metric dimensions of hierarchical product graphs, introducing a new rooted metric dimension parameter and deriving formulas and bounds for these dimensions.

Contribution

It introduces the rooted metric dimension for rooted graphs and establishes formulas and bounds for the metric and fractional metric dimensions of hierarchical product graphs.

Findings

Exact formula for metric dimension when $G_1$ is not a path.

Tight inequalities for the case when $G_1$ is a path.

Results extend to fractional metric dimension.

Abstract

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.