Bounded degree and planar spectra

TL;DR

This paper explores the complexity of finite spectra when models are restricted to be planar or have bounded degree, revealing that these restrictions still encompass a large portion of the complexity class NE.

Contribution

It demonstrates that significant fragments of NE are captured by planar and bounded-degree spectra, while also establishing upper bounds on their expressiveness.

Findings

Significant fragments of NE are included in planar and bounded-degree spectra.

Not all sets in NE can be realized as such spectra.

The class of these spectra is surprisingly rich despite restrictions.

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.