Type of Document Dissertation Author Gaslowitz, Joshua Zachary Author's Email Address j.zachary.gaslowitz@vanderbilt.edu URN etd-03262018-125736 Title Characterizations of Graphs Without Certain Small Minors Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Mark Ellingham Committee Chair Bruce Hughes Committee Member Dong Ye Committee Member Jerry Spinrad Committee Member Mike Mihalik Committee Member Paul Edelman Committee Member Keywords

- restricted minor
- graph minor
Date of Defense 2018-03-19 Availability unrestricted AbstractThis dissertation contributes new results about minor-restricted families of graphs to the field of structural graph theory. A graph G contains a graph H as a minor if H can be formed from G through a sequence of vertex deletions, edge deletions, and edge contractions. Graph minors are of interest in part because they often serve as obstructions to having important properties. For example, Wagner's Theorem characterizes graphs that can be embedded in the plane as exactly those which contain neither K_{5}nor K_{3,3}as minors.It is known that the set of graphs embeddable in any fixed surface, and indeed each minor-closed set of graphs, has a similar characterization in terms of a finite list of forbidden minors --- though finding this list is, in general, very difficult. The related problem of fixing a graph H and describing the family of H-minor-free graphs is also difficult, although a very rough structural description that applies to any H is known. A precise characterization of such families has been found for several specific minors, sometimes with an additional constraint on the connectivity of the graphs. Our work provides a characterization and enumeration of 4-connected K

_{2,5}-minor-free graphs and a characterization of planar 4-connected DW_{6}-minor-free graphs, where DW_{6}is the join of C_{6}with two independent vertices.Files

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