Doctor of Philosophy (PhD)
A graph is almost series-parallel if there is some edge that one can add to the graph and then contract out to leave a series-parallel graph, that is, a graph with no K4-minor. In this dissertation, we find the full list of excluded minors for the class of graphs that are almost series-parallel. We also obtain the corresponding result for the class of graphs such that uncontracting an edge and then deleting the uncontracted edge produces a series-parallel graph.
A notable feature of a 3-connected almost series-parallel graph is that it has two vertices whose removal leaves a tree. This motivates consideration of those graphs for which there are two vertices whose removal is cycle-free. We find the full list of excluded minors for the class of graphs that have a set of at most two vertices whose removal is cycle-free.
Document Availability at the Time of Submission
Release the entire work immediately for access worldwide.
Warshauer, Lisa, "Some classes of graphs that are nearly cycle-free" (2011). LSU Doctoral Dissertations. 1533.
Oxley, James G.