A graph G contains another graph H as an immersion if H can be obtained from a subgraph of G by splitting off edges and removing isolated vertices. There is an obvious necessary degree condition for the immersion containment: if G contains H as an immersion, then for every integer k, the number of vertices of degree at least k in G is at least the number of vertices of degree at least k in H. In this paper, we prove that this obvious necessary condition is “nearly” sufficient for graphs with no edge-cut of order 3: for every graph H, every H-immersion free graph with no edge-cut of order 3 can be obtained by an edge-sum of graphs, where each of the summands is obtained from a graph violating the obvious degree condition by adding a bounded number of edges. The condition for having no edge-cut of order 3 is necessary. A simple application of this theorem shows that for every graph H of maximum degree $d\ge 4$, there exists an integer c such that for every positive integer m, there are at most $c^m$ unlabeled d-edge-connected H-immersion free m-edge graphs with no isolated vertex, while there are superexponentially many unlabeled $(d-1)$-edge-connected H-immersion free m-edge graphs with no isolated vertex. Our structure theorem will be applied in a forthcoming paper about determining the clustered chromatic number of the class of H-immersion free graphs.

一个图G包含另一个图H作为浸入，如果H可以从G的子图中通过分裂边和移除孤立的顶点获得。浸没包含有一个明显的必要度条件：如果G包含H作为浸没，那么对于每个整数k，G中度数至少为k的顶点数至少是度数至少为k的顶点数。H。在本文中，我们证明了对于没有 3 阶边割的图，这个明显的必要条件“几乎”充分：对于每个图H，每个没有 3 阶边割的H浸没自由图都可以通过图的边和来获得，其中每个被加数是通过添加有界数量的边从违反明显度条件的图获得的。没有3阶切边的条件是必要的。该定理的一个简单应用表明，对于每个最大度数的图H$d\ge 4$, 存在一个整数c使得对于每个正整数m , 至多有$C^{米}$没有孤立顶点的未标记的d边连接的H沉浸自由m边图，而未标记的超指数$(d-1)$-edge-connected H -immersion free m -edge 图，没有孤立的顶点。我们的结构定理将在即将发表的一篇关于确定H浸没自由图类的聚类色数的论文中应用。