Chord Diagrams and Gauss Codes for Graphs

TL;DR

This paper extends the concept of chord diagrams from circles to general planar graphs, introduces Gauss codes for graph immersions, and provides algorithms to determine their realizability, advancing the study of spatial graphs.

Contribution

It defines chord diagrams for planar graph embeddings, proves foundational results, and develops algorithms for Gauss code realizability in graph immersions.

Findings

Chord diagrams are generalized to planar graphs.

Basic properties of these diagrams are established.

Algorithms for Gauss code realizability are introduced.

Abstract

Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general graphs have not been studied, and are potentially equally valuable in the study of spatial graphs. We will define chord diagrams for planar embeddings of planar graphs and their intersection graphs, and prove some basic results. Then, as an application, we will introduce Gauss codes for immersions of graphs in the plane and give algorithms to determine whether a particular crossing sequence is realizable as the Gauss code of an immersed graph.