# Some Results on Minors for Graphs and Matroids.

1991

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

Mathematics

James G. Oxley

## Abstract

This dissertation solves some problems relating to the theory of graphs. The first type of problem considered concerns the structure of various classes of graphs which arise naturally from outerplanar graphs. These problems are motivated by Chartrand and Harary's well-known characterization of outerplanar graphs. This theorem states that K\$\sb4\$ and K\$\sb{2,3}\$ are the only non-outerplanar graphs for which both G\$\\\$e, the deletion of the edge e from the graph G, and G/e, the contraction of the edge e, are outerplanar for all edges e of G. Following Gubser's characterization of almost-planar graphs, we begin our study of graphs related to outerplanar graphs by characterizing the non-outerplanar graphs for which G\$\\\$e or G/e is outerplanar. We call these graphs almost-outerplanar (or 1-outerplanar). We then consider the corresponding problem for almost-outerplanar graphs and characterize the graphs G that are not almost-outerplanar such that G\$\\\$e or G/e is almost-outerplanar or outerplanar for every edge e of G. We end our study of graphs arising from outerplanar graphs by relaxing Chartrand and Harary's condition characterizing outerplanarity in a different way. This time we describe the non-outerplanar graphs G for which G\$\\\$e and G/e are outerplanar for at least one edge e. The second problem we solve is motivated by Hartvigsen and Zemel's characterization of graphs having the property that every circuit basis is fundamental. This theorem states that a graph has every circuit basis fundamental if and only if the graph has no minor isomorphic to one of five graphs. We consider the corresponding problem for binary matroids. We show that, in general, the class of binary matroids for which every circuit basis is fundamental is not closed under the taking of minors. However, this class is closed under the taking of series-minors. We also described some general properties of this class of matroids. We end this chapter by extending Hartvigsen and Zemel's result to the class of regular matroids.

101