(c-)And(1): a new graph model - more than intersection, more than geometric

We introduce a new graph model. In this model, vertices are points spread in a metric space and each vertex is assigned to a box in the space. Two vertices are connected by an edge if and only if its respective boxes contain the opposite vertex. We focus our study to the simplest case in which vertices are spread in a one dimensional euclidean metric space. We give both, a combinatorial and a simple intersection characterization of the model. Based on these representations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., interval graphs, outerplanar) for which we construct representation algorithms.