Minimal prime ages, words and permutation graphs Extended abstract

TL;DR

This paper classifies minimal prime hereditary classes of finite graphs, showing they are well quasi ordered, many are permutation graphs, and linking them to recurrent 0-1 words, advancing understanding of graph structure complexity.

Contribution

It provides a complete characterization of minimal prime hereditary classes, including their well quasi ordering properties and connections to permutation graphs and recurrent words.

Findings

Uncountably many minimal prime classes exist.

Eleven classes remain well quasi ordered with added labels.

Most characterized classes are permutation graphs.

Abstract

This paper is a contribution to the study of hereditary classes of finite graphs. We classify these classes according to the number of prime structures they contain. We consider such classes that are \emph{minimal prime}: classes that contain infinitely many primes but every proper hereditary subclass contains only finitely many primes. We give a complete characterization of such classes. In fact, each one of these classes is a well quasi ordered age and there are uncountably many of them. Eleven of these ages remain well quasi ordered when labels in a well quasi ordering are added. Among the remaining ones, countably many remain well quasi ordered when one label is added. Except for six examples, members of these ages we characterize are permutation graphs. In fact, every age which is not among the eleven ones is the age of a graph associated to a uniformly recurrent $0$-$1$ word on the integers. A characterization of minimal prime classes of posets and bichains is also provided.