Splitter theorems for 3- and 4-regular graphs

Abstract

Let g be a class of graphs and ≤ be a graph containment relation. A splitter theorem for g under ≤ is a result that claims the existence of a set O of graph operations such that if G and H are in g and H≤G with G≠H, then there is a decreasing sequence of graphs from G to H, say G=G0≥G1≥G2...Gt=H, all intermediate graphs are in g, and each Gi can be obtained from Gi-1 by applying a single operation in O. The classes of graphs that we consider are either 3-regular or 4-regular that have various connectivity and girth constraints. The graph containment relation we are going to consider is the immersion relation. It is worth while to point out that, for 3-regular graphs, this relation is equivalent to the topological minor relation. We will also look for the minimal graphs in each family. By combining these results with the corresponding splitter theorems, we will have several generating theorems. In Chapter 4, we investigate 4-regular planar graphs. We will see that planarity makes the problem more complicated than in the previous cases. In Section 4.5, we will prove that our results in Chapter 4 are the best possible if we only allow finitely many graph operations.