Genetic Theory for Cubic Graphs

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.

A Study on Parthenogenesis of Petersen Graph

“…If an edge which is predicted to be n-cracker is omitted, but in fact it does not make a graph is not connected, then still it is named as common cutset. N-cracker only applies on Cubic Graph [5].…”

“…An n-cracker with n ∈ {1, 2, 3} is called cubic cracker. Moreover, if a Cubic Graph does not have any cubic cracker then that graph is called gene [5]. …”

“…Similar to breeding, parthenogenesis process in cubic graph can be divided into three types [5]: 1) Parthenogenesis Type I: This happens on cubic graph with 1-cracker, by inserting diamond graph into cubic graph by cutting the segment of graph's cutset. Type I of parthenogenesis will produce a single graph with a diamond graph on the edge connectivity.…”

“…Principles and rules of genetics have been investigated by [1]. Then, [5] [5] stated that genetic processes in Cubic Graph are classified into two: the breeding and parthenogenesis.…”

A new heuristic for detecting non-Hamiltonicity in cubic graphs

“…Specifically, the structures that we search for are those that identify "brittle points" in the graph. Recently, in Baniasadi et al [2,3], it was demonstrated that the set of all connected cubic graphs can be separated into two disjoint subsets, namely genes and descendants. The key distinction between these two subsets is the presence (or absence) of special edge cut sets known as cubic crackers.…”