On grounded L-graphs and their relatives

TL;DR

This paper characterizes the class of Grounded-L graphs through vertex orderings and forbidden patterns, compares it with related graph classes, and demonstrates their distinctness through constructions.

Contribution

It provides an equivalent characterization of Grounded-L graphs via vertex orderings and forbidden patterns, and clarifies their relationship with similar intersection graph classes.

Findings

Grounded-L graphs are characterized by specific vertex orderings with forbidden patterns.

Grounded-L graphs are distinct from grounded segment, monotone L-, and outer-1-string graphs.

The classes compared only have trivial or known inclusions, showing their fundamental differences.

Abstract

We consider the graph class Grounded-L corresponding to graphs that admit an intersection representation by L-shaped curves, where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that Grounded-L graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns. We also compare this class to related intersection classes, such as the grounded segment graphs, the monotone L-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give constructions showing that these classes are all distinct and satisfy only trivial or previously known inclusions.