  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.
