  In this document, we study the scope of the following graph model: each
vertex is assigned to a box in a metric space and to a representative element
that belongs to that box. Two vertices are connected by an edge if and only if
its respective boxes contain the opposite representative element. We focus our
study on the case where boxes (and therefore representative elements)
associated to vertices are spread in the Euclidean line. We give both, a
combinatorial and an intersection characterization of the model. Based on these
characterizations, we determine graph families that contain the model (e. g.,
boxicity 2 graphs) and others that the new model contains (e. g., rooted
directed path). We also study the particular case where each representative
element is the center of its respective box. In this particular case, we
provide constructive representations for interval, block and outerplanar
graphs. Finally, we show that the general and the particular model are not
equivalent by constructing a graph family that separates the two cases.

Discrete Mathematics;Combinatorics