How not to characterize planar-emulable graphs

Abstract

We investigate the question of which graphs have planar emulators (a locally-surjective homomorphism from some finite planar graph) - a problem raised in Fellows' thesis (1985) and conceptually related to the better known planar cover conjecture by Negami (1986). For over two decades, the planar emulator problem lived poorly in a shadow of Negami's conjecture - which is still open - as the two were considered equivalent. But, in the end of 2008, a surprising construction by Rieck and Yamashita falsified the natural "planar emulator conjecture", and thus opened a whole new research field. We present further results and constructions which show how far the planar-emulability concept is from planar-coverability, and that the traditional idea of likening it to projective embeddability is actually very out-of-place. We also present several positive partial characterizations of planar-emulable graphs. © 2011 Springer-Verlag.