The covering graph of a lattice which is contained in an ordered set with a planar covering graph is itself planar. AMS subject classifications (1980). 06A10

Objective(s): We recently developed a new uptake index method for 123I-metaiodobenzylguanidine (123I-MIBG) heart uptake measurements by using planar images (radioisotope angiography and planar image) for the diagnosis of Lewy body disease (LBD), including Parkinson’s disease (PD) and dementia with Lewy bodies(DLB). However, the diagnostic accuracy of the uptake index was approximately equal to ...

In 1988, Seiya Negami published a conjecture stating that a graph G has a finite planar cover (i.e. a homomorphism from some planar graph onto G which maps the vertex neighbourhoods bijectively) if and only if G embeds in the projective plane. Though the ”if” direction is easy, and over ten related research papers have been published during the past 20 years of investigation, this beautiful con...

A graph G has a planar cover if there exists a planar graph H , and a homomorphism φ : H → G that maps the neighbours of each vertex bijectively. Each graph that has an embedding in the projective plane also has a finite planar cover. Negami conjectured the converse in 1988. This conjecture holds as long as no minor-minimal non-projective graph has a finite planar cover. From the list there rem...