All minor-minimal apex obstructions with connectivity two

TL;DR

This paper identifies all 133 minor-minimal non-apex graphs with connectivity two, advancing understanding of apex obstructions within graph theory.

Contribution

It provides the complete list of minor-minimal non-apex graphs with connectivity two, solving a long-standing open problem.

Findings

Identified 133 minor-minimal non-apex graphs with connectivity two.

Characterized the structure of apex obstructions with connectivity two.

Contributed to the classification of apex graphs in graph theory.

Abstract

A graph is an apex graph if it contains a vertex whose deletion leaves a planar graph. The family of apex graphs is minor-closed and so it is characterized by a finite list of minor-minimal non-members. The long-standing problem of determining this finite list of apex obstructions remains open. This paper determines the 133 minor-minimal, non-apex graphs that have connectivity two.