Well-Quasi-Ordering Eulerian Digraphs: Bounded Carving Width

2026-05-08

AI Summary

This study investigates the well-quasi-ordering of Eulerian digraphs under strong immersion when cutwidth is bounded. By introducing vertex labeling, edge-ordering constraints, and path-routing restrictions, the work establishes the first meta-theorem framework for strong immersion well-quasi-ordering tailored to Eulerian digraphs. Combining immersion theory, well-quasi-ordering principles, and analyses of treewidth and cutwidth, the authors prove that Eulerian digraphs with bounded cutwidth—equivalently, those with both bounded degree and bounded treewidth—are well-quasi-ordered under strong immersion. However, if vertex degrees are unbounded, strong immersion fails to preserve well-quasi-ordering even when treewidth is at most two, although weak immersion may still hold under certain conditions. These results highlight the critical role of degree bounds in ensuring strong immersion well-quasi-ordering and precisely delineate the structural constraints governing this property in Eulerian digraphs.

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.

Problem

• well-quasi-ordering
• Eulerian digraphs
• strong immersion
• carving width
• bounded degree

Innovation

• well-quasi-ordering
• Eulerian digraphs
• strong immersion
• carving width
• meta theorem