Containment Graphs, Posets, and Related Classes of Graphs
Martin Charles Golumbic, Edward R. Scheinerman

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.
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 be a family of nonempty sets. We call a (simple, finite) graph G = (V, E) a -containment graph provided one can assign to each vertex a set such that if and only if or . Similarly, we call a (strict) partially ordered set a -containment poset if to each we can assign a set such that if and only if . Obviously, is the comparability graph of . We give some basic results on containment graphs and investigate the containment graphs of iso-oriented boxes in -space. We present a characterization of those classes of posets and graphs that have containment representations by sets of a specific type,…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Limits and Structures in Graph Theory
