Bounded degree and planar spectra

Abstract

The finite spectrum of a first-order sentence is the set of positive integers that are the sizes of its models. The class of finite spectra is known to be the same as the complexity class NE. We consider the spectra obtained by limiting models to be either planar (in the graph-theoretic sense) or by bounding the degree of elements. We show that the class of such spectra is still surprisingly rich by establishing that significant fragments of NE are included among them. At the same time, we establish non-trivial upper bounds showing that not all sets in NE are obtained as planar or bounded-degree spectra.

Dynamic Distances in Hyperbolic Graphs

“…Graphs with structure typical to hyperbolic geometry appear in computer science; examples include skip lists which essentially use randomly generated hyperbolic graphs to construct an efficient dictionary, as well as Fenwick trees, quadtrees and octrees which are essentially based on the binary tiling and its higher dimensional variants. The paper [10], where all the basic constructions are essentially hyperbolic graphs. Understanding hyperbolic graphs may lead to new discoveries in computer science.…”

Hyperbolic triangulations and discrete random graphs

“…It is possible to avoid this singularity by changing our construction a bit, by making R d (G) into infinite paths (horocycles) [1]. Another possible change to our construction is to connect the last element of R d (G) with the adjacent element of R d+1 (G), thus putting all the vertices of G in a single spiral [9].…”

“…Apart from visualizations, hyperbolic triangulations have been used to create more efficient self-organizing maps (HSOMs) [21]. They also arise naturally when working with bounded degree planar graphs; for example, many constructions in [9] are Gromov hyperbolic graphs. Hyperbolic geometry is useful in mathematical art and game design [14].…”