Excluded minors for the Klein Bottle I. Low connectivity case

TL;DR

This paper characterizes the minimal obstructions for embedding graphs into the Klein bottle with low connectivity, providing a complete list of such obstructions for connectivity 2, revealing fewer than expected obstructions.

Contribution

It introduces a detailed structural analysis of critical graphs with a 2-vertex-cut for Klein bottle embeddability, including classification of building blocks and a complete list of obstructions.

Findings

Complete list of connectivity-2 obstructions for Klein bottle embedding.

Fewer excluded minors than previously anticipated.

Structural theorem for graphs with 2-vertex-cuts in this context.

Abstract

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.