Planar Doodles: Their Properties, Codes and Classification

TL;DR

This paper studies planar doodles as 4-valent graphs, classifies them into prime types, introduces a coding method for their enumeration, and explores their connection to twin groups and Hamiltonian circuits.

Contribution

It introduces a classification scheme for planar doodles, a doodle coding method, and links doodle properties to twin groups and Hamiltonian circuits.

Findings

Classification into prime and super prime doodles

Development of a doodle coding method for enumeration

Super prime doodles possess Hamiltonian circuits

Abstract

We present those properties of planar doodles, especially when regarded as 4-valent graphs, that enable us to classify them into {\it prime} and {\it super prime} doodles by analogy to a knot sum. We describe a method for partially characterising a doodle diagram by a {\it doodle code} that describes the complementary regions of the diagram and use that code to enumerate all possible prime and super prime doodle diagrams via their dual graph. In addition we explore the relationship between planar doodles and twin groups, and note that a theorem of Tutte means that super prime doodles have a Hamiltonian circuit. We hope to expand upon this last point in a follow-up paper.