# Containment Graphs, Posets, and Related Classes of Graphs

## TL;DR

This paper introduces the concept of containment graphs and posets based on set inclusion, explores their properties, characterizations, and extensions to various classes of graphs, including geometric and intersection classes.

## Contribution

It defines containment graphs and posets, characterizes classes with specific containment representations, and extends results to geometric and intersection graph classes.

## Key findings

- Containment graphs are characterized by set inclusion relations.
- Containment classes for specific set types are characterized.
- Nonexistence of a universal characterization for strong containment classes.

## Abstract

In this paper, we introduce the notion of the containment graph of a family of sets and containment classes of graphs and posets. Let $Z$ be a family of nonempty sets. We call a (simple, finite) graph G = (V, E) a $Z$-containment graph provided one can assign to each vertex $v_i \in V $ a set $S_i \in Z$ such that $v_i v_j \in E$ if and only if $S_i \subset S_j$ or $S_j \subset S_i$ . Similarly, we call a (strict) partially ordered set $P = (V, <)$ a $Z$-containment poset if to each $v_i \in V $ we can assign a set $S_i \in Z$ such that $v_i < v_j$ if and only if $S_i \subset S_j$. Obviously, $G$ is the comparability graph of $P$. We give some basic results on containment graphs and investigate the containment graphs of iso-oriented boxes in $d$-space. We present a characterization of those classes of posets and graphs that have containment representations by sets of a specific type, and we extend our results to ``injective'' containment classes. After that we discuss similar characterizations for intersection, overlap, and disjointedness classes of graphs. Finally, in the last section we discuss the nonexistence of a characterization theorem for ``strong'' containment classes of graphs.

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Source: https://tomesphere.com/paper/1907.07414