#### Abstract

The *direct sum* of a finite number of graph classes H_1, ..., H_k is defined as the set of all graphs formed by taking the union of graphs from each of the H_i. The *join* of these graph classes is similarly defined as the set of all graphs formed by taking the join of graphs from each of the H_i. In this paper we show that if each H_i has a forbidden subgraph characterization then the direct sum and join of these H_i also have forbidden subgraph characterizations. We provide various results which in many cases allow us to exactly determine the minimal forbidden subgraphs for such characterizations. As we develop these results we are led to study the minimal graphs which are *universal* over a given list of graphs, or those which contain each graph in the list as an induced subgraph. As a direct application of our results we give an alternate proof of a theorem of Barrett and Loewy concerning a forbidden subgraph characterization problem.

#### Degree

MS

#### College and Department

Physical and Mathematical Sciences; Mathematics

#### Rights

http://lib.byu.edu/about/copyright/

#### BYU ScholarsArchive Citation

Barrus, Michael D., "A Forbidden Subgraph Characterization Problem and a Minimal-Element Subset of Universal Graph Classes" (2004). *All Theses and Dissertations*. 125.

http://scholarsarchive.byu.edu/etd/125

#### Date Submitted

2004-03-17

#### Document Type

Thesis

#### Handle

http://hdl.lib.byu.edu/1877/etd374

#### Keywords

forbidden, subgraph, graph theory, universal