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.

翻译：我们证明了每个具有有界剖分宽度（等价于有界度和树宽）的欧拉有向图类在强浸入关系下是良拟序的。实际上，我们证明了一个更强的结论：每个具有有界剖分宽度且每个顶点额外标记自一个良拟序、固定其邻接边的线序、并可能对浸入路径穿越顶点的方式施加进一步限制的欧拉有向图类，在适当定义的强浸入概念下是良拟序的。为此，我们建立了一个似乎适用于通过（强）浸入证明欧拉有向图类良拟序性的框架，并给出了该方向上的首个元定理。作为补充，我们观察到具有无界度的欧拉有向图类在强浸入下不是良拟序的，即使假定该类树宽至多为2。最后，我们得出结论性的二分结果：对于一类非常受限的无界度欧拉有向图，它不在强浸入下良拟序，但在弱浸入下良拟序。