Genetic Theory for Cubic Graphs

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

Wu: A polynomial algorithm for cyclic edge connectivity of cubic graphs

In this paper, we develop a polynomial time algorithm to find out all the minilnum cyclic edge cutsets of a 3-regular graph, and therefore to determine the cyclic edge connectivity of a cubic graph. The algorithm is recursive, with complexity bounded by O(n31og2 n). The algorithm shows that the number of mini~um cyclic edge cut sets of a 3-regular graph G is polynornial in v ( G) and that the minimum cyclic edge cutsets can be found in polynomial time, and so the cyclic edge connectivity of G can be calculated. O