Graph Minors and Minimum Degree

Let Dk be the class of graphs for which every minor has minimum degree at most k. Then Dk is closed under taking minors. By the Robertson-Seymour graph minor theorem, Dk is characterised by a finite family of minor-minimal forbidden graphs, which we denote by D̂k. This paper discusses D̂k and related topics. We obtain four main results: 1. We prove that every (k + 1)-regular graph with less than 4 3(k + 2) vertices is in D̂k, and this bound is best possible. 2. We characterise the graphs in D̂k+1 that can be obtained from a graph in D̂k by adding one new vertex. 3. For k 6 3 every graph in D̂k is (k + 1)-connected, but for large k, we exhibit graphs in D̂k with connectivity 1. In fact, we construct graphs in Dk with arbitrary block structure. 4. We characterise the complete multipartite graphs in D̂k, and prove analogous characterisations with minimum degree replaced by connectivity, treewidth, or pathwidth. D.W. is supported by a QEII Research Fellowship from the Australian Research Council. An extended abstract of this paper was published in: Proc. Topological & Geometric Graph Theory (TGGT ’08), Electronic Notes in Discrete Mathematics 31:79-83, 2008. the electronic journal of combinatorics 17 (2010), #R151 1