  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.

Combinatorics