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A Characterization Of Almost All Minimal Not Nearly Planar Graphs, Kwang Ju Choi
A Characterization Of Almost All Minimal Not Nearly Planar Graphs, Kwang Ju Choi
LSU Doctoral Dissertations
In this dissertation, we study nearly planar graphs, that is, graphs that are edgeless or have an edge whose deletion results in a planar graph. We show that all but finitely many graphs that are not nearly planar and do not contain one particular graph have a well-understood structure based on large Möbius ladders.
Unavoidable Minors In Graphs And Matroids, Carolyn Barlow Chun
Unavoidable Minors In Graphs And Matroids, Carolyn Barlow Chun
LSU Doctoral Dissertations
It is well known that every sufficiently large connected graph G has either a vertex of high degree or a long path. If we require G to be more highly connected, then we ensure the presence of more highly structured minors. In particular, for all positive integers k, every 2-connected graph G has a series minor isomorphic to a k-edge cycle or K_{2,k}. In 1993, Oxley, Oporowski, and Thomas extended this result to 3- and internally 4-connected graphs identifying all unavoidable series minors of these classes. Loosely speaking, a series minor allows for arbitrary edge deletions but only allows edges …
Some Results On Cubic Graphs, Evan Morgan
Some Results On Cubic Graphs, Evan Morgan
LSU Doctoral Dissertations
Pursuing a question of Oxley, we investigate whether the edge set of a graph admits a bipartition so that the contraction of either partite set produces a series-parallel graph. While Oxley's question in general remains unanswered, our investigations led to two graph operations (Chapters 2 and 4) which are of independent interest. We present some partial results toward Oxley's question in Chapter 3. The central results of the dissertation involve an operation on cubic graphs called the switch; in the literature, a similar operation is known as the edge slide. In Chapter 2, the author proves that we can transform, …