Planar Doodles: Their Properties, Codes and Classification

In this paper, we present those properties of planar doodles, especially when regarded as four-valent graphs, that enable us to classify them into prime and super-prime doodles by analogy to a knot sum. We describe a method for partially characterizing a doodle diagram by a 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 and we give examples of prime doodles without a Hamiltonian circuit.