dc.contributor.author Asadi Shahmirzadi, Arash 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.identifier.uri http://hdl.handle.net/1853/45914 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. en_US 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.) 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 Dissertation en_US dc.description.degree PhD en_US dc.contributor.department Mathematics en_US dc.description.advisor Committee Chair: Thomas, Robin; Committee Member: Cook, William; Committee Member: Tetali, Prasad; Committee Member: Trotter, William; Committee Member: Yu, Xingxing en_US
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