Minimal prime ages, words and permutation graphs
Djamila Oudrar, Maurice Pouzet, Imed Zaguia

TL;DR
This paper classifies minimal prime hereditary classes of finite graphs, especially permutation graphs, based on their prime structures, and explores their properties related to well-quasi-ordering and associated words.
Contribution
It provides a complete description of minimal prime classes, characterizes their structure, and links them to permutation graphs and recurrent words, advancing the understanding of hereditary graph classes.
Findings
Uncountably many minimal prime classes exist.
Most classes are well-quasi-ordered and have finitely many bounds.
Permutation graphs play a central role in these classes.
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 description of such classes. In fact, each one of these classes is a well-quasi-ordered (w.q.o) age and there are uncountably many of them. Eleven of these ages are almost multichainable; they remain w.q.o when labels in a w.q.o are added, hence have finitely many bounds. Five ages among them are exhaustible. Among the remaining ones, only countably many remain w.q.o when one label is added, and these have finitely many bounds (except for the age of the infinite path and its complement). The others have infinitely many bounds. Except…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
Topicssemigroups and automata theory · Coding theory and cryptography · Finite Group Theory Research
