Identifier

etd-07072013-161459

Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

Archdeacon showed that the class of graphs embeddable in the projective plane is characterized by a set of 35 excluded minors. Robertson, Seymour and Thomas in an unpublished result found the excluded minors for the class of k-connected graphs embeddable on the projective plane for k = 1,2,3. We give a short proof of that result and then determine the excluded minors for the class of internally 4-connected projective graphs. Hall showed that a 3-connected graph diff_x000B_erent from K5 is planar if and only if it has K3,3 as a minor. We provide two analogous results for projective graphs. For any minor-closed class of graphs C, we say that a set of k-connected graphs E disjoint from C is a k-connected excludable set for C if all but a _x000C_finite number of k-connected graphs not in C have a minor in E. Hall's result is equivalent to saying that {K3,3} is a 3-connected excludable set for the class of planar graphs. We classify all minimal 3-connected excludable sets and fi_x000C_nd one minimal internally 4-connected excludable set for the class of projective graphs. In doing so, we also prove strong splitter theorems for 3-connected and internally 4-connected graphs that could have application to other problems of this type.

Date

2013

Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

Committee Chair

Ding, Guoli

DOI

10.31390/gradschool_dissertations.3914

Download

Share

COinS