Title:
Minor-minimal non-projective planar graphs with an internal 3-separation

dc.contributor.advisor Thomas, Robin
dc.contributor.author Asadi Shahmirzadi, Arash en_US
dc.contributor.committeeMember Cook, William
dc.contributor.committeeMember Tetali, Prasad
dc.contributor.committeeMember Trotter, William
dc.contributor.committeeMember Yu, Xingxing
dc.contributor.department Mathematics en_US
dc.date.accessioned 2013-01-17T21:59:32Z
dc.date.available 2013-01-17T21:59:32Z
dc.date.issued 2012-11-13 en_US
dc.description.abstract The property that a graph has an embedding in the projective plane is closed under taking minors. Thus by the well known Graph Minor theorem of Robertson and Seymour, there exists a finite list of minor-minimal graphs, call it L, such that a given graph G is projective planar if and only if G does not contain any graph isomorphic to a member of L as a minor. Glover, Huneke and Wang found 35 graphs in L, and Archdeacon proved that those are all the members of L, but Archdeacon's proof never appeared in any refereed journal. In this thesis we develop a modern approach and technique for finding the list L, independent of previous work. Our approach is based on conditioning on the connectivity of a member of L. Assume G is a member of L. If G is not 3-connected then the structure of G is well understood. In the case that G is 3-connected, the problem breaks down into two main cases, either G has an internal separation of order three or G is internally 4-connected. In this thesis we find the set of all 3-connected minor minimal non-projective planar graphs with an internal 3-separation. For proving our main result, we use a technique which can be considered as a variation and generalization of the method that Robertson, Seymour and Thomas used for non-planar extension of planar graphs. Using this technique, besides our main result, we also classify the set of minor minimal obstructions for a-, ac-, abc-planarity for rooted graphs. (A rooted graph (G,a,b,c) is a-planar if there exists a split of the vertex a to a' and a' in G such that the new graph G' obtained by the split has an embedding in a disk such that the vertices a', b, a', c are on the boundary of the disk in the order listed. We define b- and c-planarity analogously. We say that the rooted graph (G,a,b,c) is ab-planar if it is a-planar or b-planar, and we define abc-planarity analogously.) en_US
dc.description.degree PhD en_US
dc.identifier.uri http://hdl.handle.net/1853/45914
dc.publisher Georgia Institute of Technology en_US
dc.subject C-planar en_US
dc.subject Non-projective planar en_US
dc.subject Minor-minimal en_US
dc.subject.lcsh Algorithms
dc.subject.lcsh Graph theory
dc.subject.lcsh Graph theory
dc.subject.lcsh Geometry, Plane
dc.title Minor-minimal non-projective planar graphs with an internal 3-separation en_US
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Mathematics
relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69
Files
Original bundle
Now showing 1 - 1 of 1
Thumbnail Image
Name:
Asadi-Shahmirzadi_Arash_201212_PhD.pdf
Size:
1022.54 KB
Format:
Adobe Portable Document Format
Description: