The Equidistant Dimension of Corona Product Graphs

TL;DR

This paper explores the equidistant dimension of corona product graphs, introducing a new auxiliary graph and establishing bounds and exact values, revealing how it depends on the component graphs.

Contribution

It introduces the empty bisector graph to analyze equidistant sets in corona products and provides tight bounds and exact values for various graph families.

Findings

Derived tight bounds for the equidistant dimension of corona product graphs.

Established that the equidistant dimension depends linearly on the order of H for fixed G.

Provided exact values for classical families of graphs.

Abstract

A subset $S$ of vertices, in a connected graph $G$, is called a distance-equalizer set if for every pair of distinct vertices outside $S$, there exists a vertex in $S$ equidistant to both. The equidistant dimension, denoted by $\xi(G)$, is defined as the minimum cardinality of such sets. While several distance-based parameters have been studied for different graph products, the equidistant dimension of corona product graphs has remained unexplored. In this paper, we investigate the equidistant dimension of the corona product $G \odot H$ of two graphs $G$ and $H$. We introduce the empty bisector graph $\widehat{G}$, an auxiliary construction that relates pairs of vertices in $G$ that cannot be equidistant from any third vertex. Using this framework, we establish tight bounds on the equidistant dimension of $G \odot H$ and derive exact values for several classical families of graphs. Moreover, we show that for any fixed base graph $G$, the equidistant dimension of $G \odot H$ depends on $H$ only through its order and eventually becomes linear in $\n(H)$.