K7-minors in optimal 1-planar graphs

We discuss the existence of minors of given graphs in optimal 1-planar graphs. As our first main result, we prove that for any graph H, there exists an optimal 1-planar graph which contains H as a topological minor. Next, we consider minors of complete graphs. It is easily obtained from Mader's result (Mader, 1968) that every optimal 1-planar graph has a K6-minor. In the paper, we characterize optimal 1-planar graphs having no K7-minor.