Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the case when Euler genus is small. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained. This is the first complete result about obstructions for embeddability of graphs in the Klein bottle, and the outcome is somewhat surprising in the sense that there are considerably fewer excluded minors than expected.

研究了对于表面可嵌入性至关重要的图（最小排除次要图）。在第一部分中，我们考虑具有 2 顶点切割的图的结构，这对于欧拉属至关重要。提出了描述构建块的一般定理。这些成分称为料斗和级联，针对欧拉属较小的情况进行分类。结果，获得了用于将图嵌入到克莱因瓶中的连通性障碍2的完整列表。这是关于克莱因瓶中图形嵌入性障碍的第一个完整结果，从某种意义上说，被排除的未成年人比预期少得多，这个结果有点令人惊讶。