On the metric dimension of corona product graphs

TL;DR

This paper investigates the metric dimension of recursive corona product graphs, providing formulas for its calculation based on properties of the component graphs, especially focusing on diameter and specific graph types.

Contribution

It introduces new formulas for the metric dimension of iterated corona product graphs, extending understanding of how graph parameters influence resolving sets.

Findings

If the diameter of H is at most two, then dim(G⊙^k H) = n₁(n₂+1)^{k-1} dim(H).

For n₂ ≥ 7 and diameter of H > 5 or H is a cycle, dim(G⊙^k H) = n₁(n₂+1)^{k-1} dim(K₁⊙ H).

Provides recursive formulas for the metric dimension of complex graph constructions.

Abstract

Given a set of vertices $S=\{v_1,v_2,...,v_k\}$ of a connected graph $G$, the metric representation of a vertex $v$ of $G$ with respect to $S$ is the vector $r(v|S)=(d(v,v_1),d(v,v_2),...,d(v,v_k))$, where $d(v,v_i)$, $i\in \{1,...,k\}$ denotes the distance between $v$ and $v_i$. $S$ is a resolving set for $G$ if for every pair of vertices $u,v$ of $G$, $r(u|S)\ne r(v|S)$. The metric dimension of $G$, $dim(G)$, is the minimum cardinality of any resolving set for $G$. Let $G$ and $H$ be two graphs of order $n_1$ and $n_2$, respectively. The corona product $G\odot H$ is defined as the graph obtained from $G$ and $H$ by taking one copy of $G$ and $n_1$ copies of $H$ and joining by an edge each vertex from the $i^{th}$-copy of $H$ with the $i^{th}$-vertex of $G$. For any integer $k\ge 2$, we define the graph $G\odot^k H$ recursively from $G\odot H$ as $G\odot^k H=(G\odot^{k-1} H)\odot H$. We give several results on the metric dimension of $G\odot^k H$. For instance, we show that given two connected graphs $G$ and $H$ of order $n_1\ge 2$ and $n_2\ge 2$, respectively, if the diameter of $H$ is at most two, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(H)$. Moreover, if $n_2\ge 7$ and the diameter of $H$ is greater than five or $H$ is a cycle graph, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(K_1\odot H).$