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 donít have the reference for this paper)


Submitted by: Dan Archdeacon, Dept. of Math. and Stat., University of Vermont, Burlington, VT 05401-1455 USA (with thanks to [BFKRS])

Send comments to dan.archdeacon@uvm.edu

December 2003