Doctor of Philosophy (PhD)
A series-parallel graph can be built from a single-edge graph by a sequence of series and parallel extensions. The class of such graphs coincides with the class of graphs that do not have the complete graph K4 as a minor. This dissertation considers a class M1 of graphs that are close to being series-parallel. In particular, every member of the class has the property that one can obtain a series-parallel graph by adding a new edge and contracting it out, or by splitting a vertex into two vertices whose neighbor sets partition the neighbor set of the original vertex. The class M1 is minor-closed. The goal of this dissertation is to show that M1 has exactly twelve excluded minors, including K5, the cube, and the octahedron.
Fontaine, Victoria, "Characterizations of Some Classes of Graphs That Are Nearly Series-Parallel" (2017). LSU Doctoral Dissertations. 4179.