(c-)AND: A new graph model

TL;DR

This paper introduces a new graph model based on boxes and representatives in a metric space, providing characterizations, inclusions, and specific cases, especially on the Euclidean line, with applications to various graph families.

Contribution

The paper defines a novel graph model involving boxes and representatives, characterizes it combinatorially and intersection-wise, and explores its relation to known graph classes, including special cases and separations.

Findings

Characterization of the model via combinatorial and intersection methods.

Identification of graph families contained within the model, such as boxicity 2 graphs.

Constructive representations for interval, block, and outerplanar graphs in the special case.

Abstract

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.