Title:
New Tools and Results in Graph Structure Theory

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dc.contributor.advisor Thomas, Robin
dc.contributor.author Hegde, Rajneesh en_US
dc.contributor.committeeMember Bill Cook
dc.contributor.committeeMember Tetali, Prasad
dc.contributor.committeeMember Richard Duke
dc.contributor.committeeMember Yu, Xingxing
dc.contributor.department Mathematics en_US
dc.date.accessioned 2006-06-09T18:09:37Z
dc.date.available 2006-06-09T18:09:37Z
dc.date.issued 2006-03-30 en_US
dc.description.abstract We first prove a ``non-embeddable extensions' theorem for polyhedral graph embeddings. Let G be a ``weakly 4-connected' planar graph. We describe a set of constructions that produce a finite list of non-planar graphs, each having a minor isomorphic to G, such that every non-planar weakly 4-connected graph H that has a minor isomorphic to G has a minor isomorphic to one of the graphs in the list. The theorem is more general and applies in particular to polyhedral embeddings in any surface. We discuss an approach to proving Jorgensen's conjecture, which states that if G is a 6-connected graph with no K_6 minor, then it is apex, that is, it has a vertex v such that deleting v yields a planar graph. We relax the condition of 6-connectivity, and prove Jorgensen's conjecture for a certain sub-class of these graphs. We prove that every graph embedded in the Klein bottle with representativity at least 4 has a K_6 minor. Also, we prove that every ``locally 5-connected' triangulation of the torus, with one exception, has a K_6 minor. (Local 5-connectivity is a natural notion of local connectivity for a surface embedding.) The above theorem uses a locally 5-connected version of the well-known splitter theorem for triangulations of any surface. We conclude with a theoretically optimal algorithm for the following graph connectivity problem. A shredder in an undirected graph is a set of vertices whose removal results in at least three components. A 3-shredder is a shredder of size three. We present an algorithm that, given a 3-connected graph, finds its 3-shredders in time proportional to the number of vertices and edges, when implemented on a RAM (random access machine). en_US
dc.description.degree Ph.D. en_US
dc.format.extent 1137497 bytes
dc.format.mimetype application/pdf
dc.identifier.uri http://hdl.handle.net/1853/10481
dc.language.iso en_US
dc.publisher Georgia Institute of Technology en_US
dc.subject Graph algorithms en_US
dc.subject Graph minors
dc.subject Graph theory
dc.title New Tools and Results in Graph Structure Theory 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
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