A chain theorem for 3 \u3csup\u3e+\u3c/sup\u3e-connected graphs

A 3-connected graph is called 3 +-connected if it has no 3-separation that separates a large fan or K 3,n from the rest of the graph. It is proved in this paper that except for K 4, every 3 +-connected graph has a 3 +-connected proper minor that is at most two edges away from the original graph. This result is used to characterize Q-minor-free graphs, where Q is obtained from the cube by contracting an edge. © 2012 Society for Industrial and Applied Mathematics

Excluding a small minor

There are sixteen 3-connected graphs on eleven or fewer edges. For each of these graphs H we discuss the structure of graphs that do not contain a minor isomorphic to H. © 2012 Elsevier B.V. All rights reserved

Connectivity, tree-decompositions and unavoidable-minors

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