Cascades and Obstructions of Low Connectivity for Embedding Graphs into the Klein Bottle

TL;DR

This paper characterizes the structure of certain low-connectivity graphs critical for Klein bottle embeddings, providing a classification of obstructions and building blocks like hoppers and cascades.

Contribution

It introduces a general theorem on the structure of these graphs and provides a complete list of obstructions for embedding into the Klein bottle.

Findings

Classification of hoppers and cascades for small Euler genus

Complete list of connectivity 2 obstructions for Klein bottle embedding

Structural theorem describing building blocks of critical graphs

Abstract

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.