Martin Derka — Page 2

arXiv2013

Dihedral angles and orthogonal polyhedra

Consider an orthogonal polyhedron, i.e., a polyhedron where (at least after a suitable rotation) all faces are perpendicular to a coordinate axis, and hence all edges are parallel to a coordinate axis. Clearly, any facial angle and any dihedral angle is a multiple of $π/2$. In this note we explore the converse: if the facial and/or dihedral angles are all multiples of $π/2$, is the polyhedron necessarily orthogonal? The case of facial angles was answered previously. In this note we show that if both the facial and dihedral angles are multiples of $π/2$ then the polyhedron is orthogonal (presuming connectivity), and we give examples to show that the condition for dihedral angles alone does not suffice.

cs.CG↗

arXiv2011

How Not to Characterize Planar-emulable Graphs

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.

cs.DM↗