Understanding lettericity I: a structural hierarchy

Abstract

Lettericity is a graph parameter introduced by Petkovšek in [16] in order to study wellquasi-orderability under the induced subgraph relation. In the world of permutations, geometric griddability was independently introduced in [1], 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 the "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.