Minor-minimal planar graphs of even branch-width

TL;DR

This paper investigates minor-minimal planar graphs with even branch-width, establishing a relationship with graphs in the projective plane and providing examples of such graphs with branch-width 6 that are constructed differently.

Contribution

It proves that certain planar double covers of minor-minimal projective plane graphs are minor-minimal with branch-width 2k and presents new examples of minor-minimal planar graphs with branch-width 6.

Findings

G is minor-minimal of branch-width 2k for certain projective plane graphs

Examples of minor-minimal planar graphs of branch-width 6 not arising from the described construction

Relationship between projective plane graphs and their planar double covers

Abstract

Let k>0 be an integer, let H be a minor-minimal graph in the projective plane such that every homotopically non-trivial closed curve intersects H at least k times, and let G be the planar double cover of H obtained by lifting G into the universal covering space of the projective plane, the sphere. We prove that G is minor-minimal of branch-width 2k. We also exhibit examples of minor-minimal planar graphs of branch-width 6 that do not arise this way.