Finite planar emulators for K4, 5-4K2 and K1, 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 K4,5 − 4K2. Archdeacon [Dan Archdeacon, Two graphswithout planar covers, J. Graph Theory, 41 (4) (2002) 318–326] showed that K4,5−4K2 does not admit a finite planar cover; thusK4,5−4K2 provides a counterexample to Fellows’ Conjecture. It is known that Negami’s Planar Cover Conjecture is true if and only if K1,2,2,2 admits no finite planar cover. We construct a finite planar emulator for K1,2,2,2. The existence of a finite planar cover for K1,2,2,2 is still open. © 2009 Elsevier Ltd. All rights reserved.