Finite planar emulators for and 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 − 4 K 2 . Archdeacon [Dan Archdeacon, Two graphs without planar covers, J. Graph Theory, 41 (4) (2002) 318–326] showed that K 4 , 5 − 4 K 2 does not admit a finite planar cover; thus K 4 , 5 − 4 K 2 provides a counterexample to Fellows’ Conjecture. 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.