## LSU Historical Dissertations and Theses

#### Title

Some Problems for Graph Minors.

1990

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

James G. Oxley

#### Abstract

This dissertation solves two problems relating to the structure of graphs. The first of these is motivated by Kuratowski's Theorem, perhaps the most famous result in graph theory. This theorem states the \$K\sb5\$ and K\$\sb{3,3}\$ are the only non-planar graphs for which both \$G\\ e\$, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the non-planar graphs for which \$G\\ e\$ or G/e is planar for all edges e of G. The second problem we solve is motivated by Tutte's wheels theorem. An immediate consequence of this theorem is that the wheel graphs are the basic building blocks for the collection of simple, 3-connected graphs. Therefore it is of interest to examine the structure of the graphs that do not have a minor isomorphic to the k-spoked wheel, \$W\sb{k}\$, for small values of k. Dirac determined that the graphs having no \$W\sb3\$-minor are the series-parallel networks. It follows easily from Tutte's wheels theorem is that \$W\sb3\$ is the only graph that has a \$W\sb3\$-minor and no \$W\sb4\$-minor. Oxley characterized the graphs that have a \$W\sb4\$-minor and no \$W\sb5\$-minor. We characterize the planar graphs that have a \$W\sb5\$-minor and no \$W\sb6\$-minor. We also determine the best-possible upper bound on the number of edges of such a graph.

143

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