Archdeacon, DanBonnington, C. Paul2009-08-282009-08-282001-09Department of Mathematics - Research Reports-470 (2001)1173-0889http://hdl.handle.net/2292/5158The {em spindle surface} $S$ is the pinched surface formed by identifying two points on the sphere. In this paper we examine cubic graphs that minimally do not embed on the spindle surface. We give the complete list of 21 cubic graphs that form the topological obstruction set in the cubic order for graphs that embed on $S$. A graph $G$ is {em nearly-planar} if there exists an edge $e$ such that $G - e $ is planar. All planar graphs are nearly-planar. A cubic obstruction for near-planarity is the same as an obstruction for embedding on the spindle surface. Hence we also give the topological obstruction set for cubic nearly-planar graphs.https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmTechnical ReportFields of Research::230000 Mathematical Sciences::230100 MathematicsThe author(s)