Arc numbers from Gauss diagrams

TL;DR

This paper characterizes planar diagrams based on their arc embeddings using chord diagrams, generalizing Taniyama's result for n=2, and provides algorithms for minimal arc embeddings and subdiagrams.

Contribution

It generalizes Taniyama's characterization to arbitrary arc numbers and introduces two algorithms for finding minimal arc embeddings and subdiagrams.

Findings

Characterization of planar diagrams with n arc embeddings

Quadratic time algorithm for minimal arc embedding

Algorithm for constructing minimal subdiagram with same arc number

Abstract

We characterize planar diagrams which may be divided into n arc embeddings in terms of their chord diagrams, generalizing a result of Taniyama for the case n = 2. Two algorithms are provided, one which finds a minimal arc embedding (in quadradic time in the number of crossings), and one which constructs a minimal subdiagram having same arc number as D.