Understanding lettericity I: a structural hierarchy

TL;DR

This paper investigates the structural hierarchy of lettericity, a graph parameter, by examining minimal obstructions and revealing connections to geometric griddability in permutations.

Contribution

It introduces an infinite collection of minimal classes of graphs with unbounded lettericity and uncovers a structural hierarchy linking lettericity and griddability.

Findings

Identified an infinite collection of minimal classes of graphs with unbounded lettericity.

Discovered a structural hierarchy connecting lettericity and geometric griddability.

Highlighted structural similarities between graph lettericity and permutation griddability.

Abstract

Lettericity is a graph parameter introduced by Petkov\v{s}ek in 2002 in order to study well-quasi-orderability under the induced subgraph relation. In the world of permutations, geometric griddability was independently introduced in 2013 by Albert, Atkinson, Bouvel, Ru\v{s}kuc and Vatter, partly as an enumerative tool. Despite their independent origins, those two notions share a connection: they highlight very similar structural features in their respective objects. The fact that those structural features arose separately on two different occasions makes them very interesting to study in their own right. In the present paper, we explore the notion of lettericity through the lens of "minimal obstructions", i.e., minimal classes of graphs of unbounded lettericity, and identify an infinite collection of such classes. We also discover an intriguing structural hierarchy that arises in the study of lettericity and that of griddability.