All Minor-Minimal Apex Obstructions with Connectivity Two

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.