Type of Document	Dissertation
Author	Kanno, Jinko
Author's Email Address	jkanno1@lsu.edu
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
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.
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