Graph Minors and Minimum Degree
  Let $\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$.   Then $\mathcal{D}_k$ is closed under taking minors.   By the Robertson-Seymour graph minor theorem, $\mathcal{D}_k$ is characterised by a finite family of minor-minimal forbidden graphs, which we denote by $\widehat{\mathcal{D}}_k$.   This paper discusses $\widehat{\mathcal{D}}_k$ and related topics. We obtain four main results:   We prove that every $(k+1)$-regular graph with less than ${4/3}(k+2)$ vertices is in $\widehat{\mathcal{D}}_k$, and this bound is best possible.   We characterise the graphs in $\widehat{\mathcal{D}}_{k+1}$ that can be obtained from a graph in $\widehat{\mathcal{D}}_k$ by adding one new vertex.   For $k\leq 3$ every graph in $\widehat{\mathcal{D}}_k$ is $(k+1)$-connected, but for large $k$, we exhibit graphs in $\widehat{\mathcal{D}}_k$ with connectivity 1. In fact, we construct graphs in $\mathcal{D}_k$ with arbitrary block structure.   We characterise the complete multipartite graphs in $\widehat{\mathcal{D}}_k$, and prove analogous characterisations with minimum degree replaced by connectivity, treewidth, or pathwidth. 
Ga\v{s}per Fijav\v{z}, David R. Wood