Equidistant dimension of Cartesian product graphs

TL;DR

This paper introduces the concept of equidistant dimension in graphs, focusing on Cartesian product graphs, and computes this measure for various specific graph families.

Contribution

It defines the equidistant dimension for connected graphs and calculates it for several classes of Cartesian product graphs, expanding understanding of graph metric properties.

Findings

Equidistant dimension computed for two-dimensional Hamming graphs

Results for hypercubes and prisms of cycles

Analysis of squared grid graphs

Abstract

Given a connected graph $G$, the equidistant dimension of $G$ represents the cardinality of the smallest set of vertices $S$ of $G$ such that for any two vertices $x,y\notin S$ there is at least one vertex in $S$ equidistant to both $x,y$ in terms of distances. In this article, we compute the equidistant dimension of some Cartesian product graphs including two-dimensional Hamming graphs, some hypercubes, prisms of cycle, and squared grid graphs.