  We introduce the notion of a centroidal locating set of a graph $G$, that is,
a set $L$ of vertices such that all vertices in $G$ are uniquely determined by
their relative distances to the vertices of $L$. A centroidal locating set of
$G$ of minimum size is called a centroidal basis, and its size is the
centroidal dimension $CD(G)$. This notion, which is related to previous
concepts, gives a new way of identifying the vertices of a graph. The
centroidal dimension of a graph $G$ is lower- and upper-bounded by the metric
dimension and twice the location-domination number of $G$, respectively. The
latter two parameters are standard and well-studied notions in the field of
graph identification.
  We show that for any graph $G$ with $n$ vertices and maximum degree at
least~2, $(1+o(1))\frac{\ln n}{\ln\ln n}\leq CD(G) \leq n-1$. We discuss the
tightness of these bounds and in particular, we characterize the set of graphs
reaching the upper bound. We then show that for graphs in which every pair of
vertices is connected via a bounded number of paths,
$CD(G)=\Omega\left(\sqrt{|E(G)|}\right)$, the bound being tight for paths and
cycles. We finally investigate the computational complexity of determining
$CD(G)$ for an input graph $G$, showing that the problem is hard and cannot
even be approximated efficiently up to a factor of $o(\log n)$. We also give an
$O\left(\sqrt{n\ln n}\right)$-approximation algorithm.
