Well-Quasi-Ordering Eulerian Digraphs: Bounded Carving Width

TL;DR

This paper establishes that classes of Eulerian directed graphs with bounded carving width are well-quasi-ordered by strong immersion, introducing a new framework and a meta theorem for such orderings.

Contribution

It proves a general meta theorem for well-quasi-ordering Eulerian directed graphs by strong immersion and explores the limits with unbounded degree classes.

Findings

Bounded carving width classes are well-quasi-ordered by strong immersion.

Unbounded degree classes are not well-quasi-ordered by strong immersion.

Restricted classes of unbounded degree graphs are well-quasi-ordered by weak immersion.

Abstract

We prove that every class of Eulerian directed graphs of bounded carving width (equivalently of bounded degree and treewidth) is well-quasi-ordered by strong immersion. In fact, we prove a stronger result, namely that every class of Eulerian directed graphs of bounded carving width, where every vertex is additionally labeled from a well-quasi-order, fixes a linear order on its incident edges, and may impose further restrictions on how the immersion is allowed to route paths through it, is well-quasi-ordered by an adequate notion of strong immersion. To this extent, we develop a framework seemingly suited to prove well-quasi-ordering for classes of Eulerian directed graphs by (strong) immersion and present a first meta theorem in that direction. We complement our results by observing that the class of Eulerian directed graphs of unbounded degree is \emph{not} well-quasi-ordered by \emph{strong} immersion, even if we assume the treewidth of the class to be at most two. We conclude with a dichotomy result, proving for a very restricted class of Eulerian directed graphs of unbounded degree that it is not well-quasi-ordered by strong immersion, but it is well-quasi-ordered by weak immersion.