Type of Document Dissertation Author Kanno, Jinko Author's Email Address firstname.lastname@example.org URN etd-0529103-123537 Title Splitter Theorems for 3- and 4-Regular Graphs Degree Doctor of Philosophy (Ph.D.) Department Mathematics Advisory Committee
Advisor Name Title Guoli Ding Committee Chair Bogdan Oporowski Committee Member Patrick Gilmer Committee Member Raymond Fabec Committee Member William Hoffman Committee Member Charles Harlow Dean's Representative Keywords
- adding handles
- girth four
- non-simple graphs
- edge reduction
Date of Defense 2003-05-01 Availability unrestricted AbstractLet 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.
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