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ItemTree-based decompositions of graphs on surfaces and applications to the traveling salesman problem(Georgia Institute of Technology, 2007-12-19) Inkmann, TorstenThe tree-width and branch-width of a graph are two well-studied examples of parameters that measure how well a given graph can be decomposed into a tree structure. In this thesis we give several results and applications concerning these concepts, in particular if the graph is embedded on a surface. In the first part of this thesis we develop a geometric description of tangles in graphs embedded on a fixed surface (tangles are the obstructions for low branch-width), generalizing a result of Robertson and Seymour. We use this result to establish a relationship between the branch-width of an embedded graph and the carving-width of an associated graph, generalizing a result for the plane of Seymour and Thomas. We also discuss how these results relate to the polynomial-time algorithm to determine the branch-width of planar graphs of Seymour and Thomas, and explain why their method does not generalize to surfaces other than the sphere. We also prove a result concerning the class C_2k of minor-minimal graphs of branch-width 2k in the plane, for an integer k at least 2. We show that applying a certain construction to a class of graphs in the projective plane yields a subclass of C_2k, but also show that not all members of C_2k arise in this way if k is at least 3. The last part of the thesis is concerned with applications of graphs of bounded tree-width to the Traveling Salesman Problem (TSP). We first show how one can solve the separation problem for comb inequalities (with an arbitrary number of teeth) in linear time if the tree-width is bounded. In the second part, we modify an algorithm of Letchford et al. using tree-decompositions to obtain a practical method for separating a different class of TSP inequalities, called simple DP constraints, and study their effectiveness for solving TSP instances.
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ItemNew Tools and Results in Graph Structure Theory(Georgia Institute of Technology, 2006-03-30) Hegde, RajneeshWe 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).
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ItemExtremal Functions for Graph Linkages and Rooted Minors(Georgia Institute of Technology, 2005-11-28) Wollan, PaulExtremal Functions for Graph Linkages and Rooted Minors Paul Wollan 137 pages Directed by: Robin Thomas A graph G is k-linked if for any 2k distinct vertices s_1,..., s_k,t_1,..., t_k there exist k vertex disjoint paths P_1,...,P_k such that the endpoints of P_i are s_i and t_i. Determining the existence of graph linkages is a classic problem in graph theory with numerous applications. In this thesis, we examine sufficient conditions that guarantee a graph to be k-linked and give the following theorems. (A) Every 2k-connected graph on n vertices with 5kn edges is k-linked. (B) Every 6-connected graph on n vertices with 5n-14 edges is 3-linked. The proof method for Theorem (A) can also be used to give an elementary proof of the weaker bound that 8kn edges suffice. Theorem (A) improves upon the previously best known bound due to Bollobas and Thomason stating that 11kn edges suffice. The edge bound in Theorem (B) is optimal in that there exist 6-connected graphs on n vertices with 5n-15 edges that are not 3-linked. The methods used prove Theorems (A) and (B) extend to a more general structure than graph linkages called rooted minors. We generalize the proof methods for Theorems (A) and (B) to find edge bounds for general rooted minors, as well as finding the optimal edge bound for a specific family of bipartite rooted minors. We conclude with two graph theoretical applications of graph linkages. The first is to the problem of determining when a small number of vertices can be used to cover all the odd cycles in a graph. The second is a simpler proof of a result of Boehme, Maharry and Mohar on complete minors in huge graphs of bounded tree-width.
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ItemMatching structure and Pfaffian orientations of graphs(Georgia Institute of Technology, 2005-07-20) Norine, SergueiThe first result of this thesis is a generation theorem for bricks. A brick is a 3-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of a decomposition procedure of Kotzig, and Lovasz and Plummer. We prove that every brick except for the Petersen graph can be generated from K_4 or the prism by repeatedly applying certain operations in such a way that all the intermediate graphs are bricks. We use this theorem to prove an exact upper bound on the number of edges in a minimal brick with given number of vertices and to prove that every minimal brick has at least three vertices of degree three. The second half of the thesis is devoted to an investigation of graphs that admit Pfaffian orientations. We prove that a graph admits a Pfaffian orientation if and only if it can be drawn in the plane in such a way that every perfect matching crosses itself even number of times. Using similar techniques, we give a new proof of a theorem of Kleitman on the parity of crossings and develop a new approach to Turan's problem of estimating crossing number of complete bipartite graphs. We further extend our methods to study k-Pfaffian graphs and generalize a theorem by Gallucio, Loebl and Tessler. Finally, we relate Pfaffian orientations and signs of edge-colorings and prove a conjecture of Goddyn that every k-edge-colorable k-regular Pfaffian graph is k-list-edge-colorable. This generalizes a theorem of Ellingham and Goddyn for planar graphs.
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ItemExtremal Functions for Contractions of Graphs(Georgia Institute of Technology, 2004-07-08) Song, ZixiaIn this dissertation, a problem related to Hadwiger's conjecture has been studied. We first proved a conjecture of Jakobsen from 1983 which states that every simple graphs on $n$ vertices and at least (11n-35)/2 edges either has a minor isomorphic to K_8 with one edge deleted or is isomorphic to a graph obtained from disjoint copies of K_{1, 2, 2, 2, 2} and/or K_7 by identifying cliques of size five. We then studied the extremal functions for complete minors. We proved that every simple graph on nge9 vertices and at least 7n-27 edges either has a minor, or is isomorphic to K_{2, 2, 2, 3, 3}, or is isomorphic to a graph obtained from disjoint copies of K_{1, 2, 2, 2, 2, 2} by identifying cliques of size six. This result extends Mader's theorem on the extremal function for K_p minors, where ple7. We discussed the possibilities of extending our methods to K_{10} and K_{11} minors. We have also found the extremal function for K_7 plus a vertex minor.
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ItemCyclically 5-connected graphs : their relevance to Tutte's 4-flow conjecture(Georgia Institute of Technology, 2000-12) Thomson, Jan McDonald
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ItemIndependent sets in bounded degree graphs(Georgia Institute of Technology, 2000-05) Heckman, Christopher Carl