This dissertation establishes two theorems which characterize the set of minimal obstructions for two classes of graphs. A minimal obstruction for a class of graphs is a graph that is not in the class but every graph that it properly contains, under some containment relation, is in the class. In Chapter 2, we provide a characterization of the class of cubic outer-planar graphs in terms of its minimal obstructions which are also called cubic obstructions in this setting. To do this, we first show that all the obstructions containing loops can be obtained from the complete set of loopless obstructions via an easily specified operation. We subsequently prove that there are only two loopless obstructions and then generate the complete list of 5 obstructions.; In Chapters 3 and 4, we provide a characterization for the more general class of outer-cylindrical graphs---those graphs that can be embedded in the plane so that there are two faces whose boundaries together contain all the vertices of the graph. In particular, in Chapter 3, we build upon the ideas of Chapter 2 by considering the operation used to generate all obstructions containing loops from those that are loopless and extend this operation to the class of outer-cylindrical graphs. We also provide a list of 26 loopless graphs and prove that each of these is a cubic obstruction for outer-cylindrical graphs. In Chapter 4, we prove that these 26 graphs are the only loopless cubic obstructions for outer-cylindrical graphs. Combining the results of Chapters 3 and 4, we then generate the complete list of 124 obstructions which is provided in an appendix.
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