Connectivity, tree-decompositions and unavoidable-minors
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The results in this thesis are steps toward bridging the gap between the handful of exact structure theorems known for minor-closed classes of graphs, and the very general, yet wildly qualitative, Graph Minors Structure Theorem. This thesis introduces a refinement of the notion of tree-width. Tree-width is a measure of how “tree-like” a graph is. Essentially, a graph is tree-like if it can be decomposed across a collection of non-crossing vertex-separations into small pieces. In our variant, which we call k-tree-width, we require that the vertex-separations each have order at most k. Tree-width and branch-width are related parameters in a graph, and we introduce a branch-width-like variant for k-tree-width. We find a dual notion, in terms of tangles, for our branch-width parameter, and we prove a generalization of Robertson and Seymour’s Grid Theorem.