This book was motivated by the notion that some of the underlying difficulty in challenging
instances of graph-based problems (e.g. the Traveling Salesman Problem) may be inherited from
simpler graphs which - in an appropriate sense - could be seen as ancestors of the given graph
instance. The authors propose a partitioning of the set of unlabeled connected cubic graphs
into two disjoint subsets named genes and descendants where the cardinality of the descendants
dominates that of the genes. The key distinction between the two subsets is the presence of
special edge cut sets called cubic crackers in the descendants. The book begins by proving
that any given descendant may be constructed by starting from a finite set of genes and
introducing the required cubic crackers through the use of six special operations called
breeding operations. It shows that each breeding operation is invertible and these inverse
operations are examined. It is therefore possible for any given descendant to identify a
family of genes that could be used to generate the descendant. The authors refer to such a
family of genes as a complete family of ancestor genes for that particular descendant. The book
proves the fundamental although quite unexpected result that any given descendant has exactly
one complete family of ancestor genes. This result indicates that the particular combination of
breeding operations used strikes the right balance between ensuring that every descendant may
be constructed while permitting only one generating set. The result that any descendant can be
constructed from a unique set of ancestor genes indicates that most of the structure in the
descendant has been in some way inherited from that very special complete family of
ancestor genes with the remaining structure induced by the breeding operations. After
establishing this the authors proceed to investigate a number of graph theoretic properties:
Hamiltonicity bipartiteness and planarity and prove results linking properties of the
descendant to those of the ancestor genes. They develop necessary (and in some cases
sufficient) conditions for a descendant to contain a property in terms of the properties of its
ancestor genes. These results motivate the development of parallelizable heuristics that first
decompose a graph into ancestor genes and then consider the genes individually. In particular
they provide such a heuristic for the Hamiltonian cycle problem. Additionally a framework for
constructing graphs with desired properties is developed which shows how many (known) graphs
that constitute counterexamples of conjectures could be easily found.
