The Equidistant Dimension of Graphs

TL;DR

This paper introduces the equidistant dimension of graphs, a new parameter measuring the minimum size of a vertex set that equalizes distances between all pairs of non-selected vertices, and explores its properties and applications.

Contribution

It defines the equidistant dimension, establishes bounds, characterizes extremal graphs, and analyzes this parameter for various graph families, linking it to other concepts like 3-AP-free sets.

Findings

Bounds related to graph order, degree, clique, and independence number.

Characterization of extremal graphs for the equidistant dimension.

Connections between equidistant dimension and 3-AP-free sets in paths and cycles.

Abstract

A subset $S$ of vertices of a connected graph $G$ is a distance-equalizer set if for every two distinct vertices $x, y \in V (G) \setminus S$ there is a vertex $w \in S$ such that the distances from $x$ and $y$ to $w$ are the same. The equidistant dimension of $G$ is the minimum cardinality of a distance-equalizer set of G. This paper is devoted to introduce this parameter and explore its properties and applications to other mathematical problems, not necessarily in the context of graph theory. Concretely, we first establish some bounds concerning the order, the maximum degree, the clique number, and the independence number, and characterize all graphs attaining some extremal values. We then study the equidistant dimension of several families of graphs (complete and complete multipartite graphs, bistars, paths, cycles, and Johnson graphs), proving that, in the case of paths and cycles, this parameter is related with 3-AP-free sets. Subsequently, we show the usefulness of distance-equalizer sets for constructing doubly resolving sets.