32 research outputs found
Finite planar emulators for K_{4,5} - 4K_2 and K_{1,2,2,2} and Fellows' Conjecture
In 1988 Fellows conjectured that if a finite, connected graph admits a finite
planar emulator, then it admits a finite planar cover. We construct a finite
planar emulator for K_{4,5} - 4K_2. Archdeacon showed that K_{4,5} - 4K_2 does
not admit a finite planar cover; thus K_{4,5} - 4K_2 provides a counterexample
to Fellows' Conjecture.
It is known that Negami's Planar Cover Conjecture is true if and only if
K_{1,2,2,2} admits no finite planar cover. We construct a finite planar
emulator for K_{1,2,2,2}. The existence of a finite planar cover for
K_{1,2,2,2} is still open.Comment: Final version. To appear in European Journal of Combinatoric
The obstructions for toroidal graphs with no 's
Forbidden minors and subdivisions for toroidal graphs are numerous. We
consider the toroidal graphs with no -subdivisions that coincide with
the toroidal graphs with no -minors. These graphs admit a unique
decomposition into planar components and have short lists of obstructions. We
provide the complete lists of four forbidden minors and eleven forbidden
subdivisions for the toroidal graphs with no 's and prove that the
lists are sufficient.Comment: 10 pages, 7 figures, revised version with additional detail