**Obstructions for the
Spindle Surface**

The *spindle surface* is formed from
the sphere by identifying two points. The two points are usually called the
north and south poles, and after identification it is called the *pinch point*. Equivalently, the spindle
surface is formed from the torus where all points on a
non-contractible cycle are identified as the single pinch point. The spindle
surface is misnamed, since it is a pseudosurface; no
neighborhood of the pinch point is homeomorphic to
the plane.

We are interested in the set of minor-minimal graphs that do not embed on
the spindle surface. Finding this set would give a forbidden minor
characterization of graphs that do not embed on the spindle surface (by
excluding these *obstructions*). For
general pseudosurfaces, the set of graphs that embed
on that pseudosurface is not closed under minors, and
in fact there are pseudosurfaces which have an
infinite of obstructions. However, graphs that embed on the spindle surface are
minor closed, and hence there is a finite obstruction set.

**Problem: **[BFKRS] Find the minor-minimal graphs that do not embed on the
spindle surface.

Bodendiek and Wagner have compiled a list of 12 obstructions under a finer partial order. Archdeacon and Bonnington [AB] have found the minimal graphs of maximum degree 3 that do not embed on the spindle surface, that is, they have the obstruction sets under the topological ordering on the set of cubic graphs.

**References:**

[AB] D. Archdeacon and C.P. Bonnington, Obstruction
sets for cubic graphs on the spindle surface, *J. of Combin. Theory Ser. B*, to appear

[BW] R. Bodendiek and K. Wagner (I don’t have the reference for this paper)

Submitted by: Dan Archdeacon, Dept. of Math. and Stat.,

Send comments to *dan.archdeacon@uvm.edu*

December 2003