How Not to Characterize Planar-emulable Graphs

TL;DR

This paper explores the properties of graphs that admit planar emulators, revealing their differences from planar covers and providing partial characterizations, thus advancing understanding of graph embeddability concepts.

Contribution

It demonstrates that planar-emulability is distinct from planar-coverability and projective embeddability, and offers new results and partial characterizations in this area.

Findings

Planar-emulability differs significantly from planar-coverability.

A construction by Rieck and Yamashita disproved the planar emulator conjecture.

Several positive partial characterizations of planar-emulable graphs are provided.

Abstract

We investigate the question of which graphs have planar emulators (a locally-surjective homomorphism from some finite planar graph) -- a problem raised already 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.