Parental Testing in Graph Theory ? By S Narjess Afzaly,∗ ANU CSIT Algorithm & Data Group, [email protected]. Supervisor: Brendan McKay.∗ Problem A sample of an irreducible graph To make a complete list of non-isomorphic simple quartic graphs with the given number of vertices. In a quartic graph, each vertex has exactly four neighbours. Applications Finding counterexamples, verifying some conjectures or refining some proofs whether in graph theory or in any other field where the problem can be modelled as a graph such as: chemistry, web mining, national security, the railway map, the electrical circuit and neural science. Our approach Canonical Construction Path (McKay 1998): A general method to recursively generate combinatorial objects in which larger objects, (Children), are constructed out of smaller ones, (Parents), by some well-defined operation, (extension), and avoiding constructing isomorphic copies at each construction step by defining a unique genuine parent for each graph. Hence a generated graph is only accepted if it has been Challenges extended from its genuine parent. • To avoid isomorphic copies when generating the list: Reduction: The inverse operation of the extension. Irreducible graph: A graph with no parent. The definition of the genuine parent can substantially affect the efficiency of the generation process. • Generating all irreducible graphs. Genuine parent 1 Has an isomorphic sibling Not genuine parent 2 2 3 4 5 5 6 5 5 6 6 7 5 8 9 6 10 10 11 10 10 11 12 10 10 11 10 10 11 12 12 12 13 10 10 11 10 13 12 Our Reduction: Deleting a vertex and adding two extra disjoint edges between its neighbours. Results extension extension extension The programs developed during this research are time and storage efficient in generating a few sub-classes of the quartic graphs. They can produce quartic graphs up to 18 vertices more than 2 times efficiently as the well-known software genreg does. Theorem: A graph is irreducible iff its vertex set can be partitioned into disjoint subsets such that each subset induces a subgraph isomorphic to one of the Future work three graphs below: • Efficient generation of triangle-free quartic graphs. • Efficient generation of bipartite quartic graphs. • Study other extensions which may lead to more efficient quartic graph generations. • The more general problem: To generate all regular graphs of degree k and girth at least g, where k ≥ 3 and g ≥ 3 are some integers. ∗Algorithms & Data, Research School of Computer Science, College of Computer Science & IT, Australian National University ANU College of Engineering & Computer Science.