Structural and Topological Graph Theory and Well-Quasi-Ordering

Submitted by admin on Wed, 11/12/2019 - 17:50
Aristotelis Chaniotis
Date of Defense
Three-member Committee
Archontia C. Giannopoulou
Stavros G. Kolliopoulos
Dimitrios M. Thilikos (Advisor)

In their Graph Minors series, Neil Robertson and Paul Seymour among other great results proved Wagner's conjecture which is today known as the Robertson and Seymour's theorem. In every step along their way to the final proof, each special case of the conjecture which they were proving was a consequence of a "structure theorem", that sufficiently general graphs contain minors or other sub-objects that are useful for the proof - or equivalently that graphs that do not contain a useful minor have a certain restricted structure, deducing that way also a useful information for the proof.

In this thesis we present -relatively short-proofs of several Robertson and Seymour's theorem's special cases, illustrating by this way the interplay between structural graph theory and graphs' well-quasi-ordering. We present also the proof of the perhaps most important special case of the Robertson and Seymour's theorem which states that embeddability in any fixed surface can be characterized by forbidding finitely many minors The later result is deduced as a well-quasi-ordering result, indicating by this way the interplay among topological graph theory and well-quasi-ordering theory. Finally, we survey results regarding the well-quasi-ordering of graphs by other than the minor graphs' relations.