Cascades and Obstructions of Low Connectivity for Embedding Graphs into   the Klein Bottle
  The structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus is studied. 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. 
Bojan Mohar and Petr \v{S}koda
math.CO