| Summary: | This 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<sub>5</sub> nor K<sub>3,3</sub> as minors.<p>
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<sub>2,5</sub>-minor-free graphs and a characterization of planar 4-connected DW<sub>6</sub>-minor-free graphs, where DW<sub>6</sub> is the join of C<sub>6</sub> with two independent vertices.
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