Minimum Degree and Graph Minors

A graph G is a minor minimal minimum degree graph (MMMD) if δ(H)<δ(G) for every proper minor H of G. We (i) determine all complete multipartite MMMD graphs and show that (ii) every small k-regular graph is a MMMD graph. Intuitively it seems that MMMD graphs are highly connected. Countering that (iii) we show that MMMD graphs may have a rich block-structure.