Refining the Characterization of Projective Graphs

• Main Author: Iverson, Perry K.
• Other Authors: Sarker, Bhaba
• Format: Others
• Language: en
• Published: LSU 2013
• Subjects: Mathematics
• Online Access: http://etd.lsu.edu/docs/available/etd-07072013-161459/

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 Robertson, Seymour and Thomas in an unpublished result found the excluded minors for the class of k-connected graphs embeddable on the projective plane for k = 1,2,3.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 different 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 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 find 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.