Genetic Theory for Cubic Graphs

TL;DR

This paper introduces a novel partitioning of unlabelled connected cubic graphs into 'genes' and 'descendants', with a focus on the role of 'crackers' in generating descendants from genes, supported by theoretical proofs and numerical experiments.

Contribution

It defines a new classification of cubic graphs into genes and descendants, and presents algorithms for their generation and reconstruction, advancing understanding of graph inheritance.

Findings

Every descendant can be generated from a finite set of genes.

Inverse operations enable reconstruction of ancestor genes.

Numerical experiments support the uniqueness conjecture of gene sets.

Abstract

We propose a partitioning of the set of unlabelled, connected cubic graphs into two disjoint subsets named genes and descendants, where the cardinality of the descendants is much larger than that of the genes. The key distinction between the two subsets is the presence of special edge cut sets, called crackers, in the descendants. We show that every descendant can be created by starting from a finite set of genes, and introducing the required crackers by special breeding operations. We prove that it is always possible to identify genes that can be used to generate any given descendant, and provide inverse operations that enable their reconstruction. A number of interesting properties of genes may be inherited by the descendant, and we therefore propose a natural algorithm that decomposes a descendant into its ancestor genes. We conjecture that each descendant can only be generated by starting with a unique set of ancestor genes. The latter is supported by numerical experiments.