Well-Quasi-Ordering by the Induced-Minor Relation

Robertson and Seymour proved Wagner\u27s Conjecture, which says that finite graphs are well-quasi-ordered by the minor relation. Their work motivates the question as to whether any class of graphs is well-quasi-ordered by other containment relations. This dissertation is concerned with a special graph containment relation, the induced-minor relation. This dissertation begins with a brief introduction to various graph containment relations and their connections with well-quasi-ordering. In the first chapter, we discuss the results about well-quasi-ordering by graph containment relations and the main problems of this dissertation. The graph theory terminology and preliminary results that will be used are presented in the next chapter. The class of graphs that is considered in this research is the class W of graphs that contain neither W4 (a wheel graph with five vertices) and K5\e (a complete graph on five vertices minus an edge) as an induced minor. Chapter 3 is devoted to studying the structure of this class of graphs. A class of graphs is well-quasi-ordered by a containment relation if it contains no infinite antichain, so infinite antichains are important. We construct in Chapter 4 an infinite antichain of W with respect to the induced minor relation and study its important properties in Chapter 5. These properties are used in determining all well-quasi-ordered subclasses of W to reach the main result of Chapter 6