$K_7$-Minors in Optimal $1$-Planar Graphs
Minisymposium: GENERAL SESSION TALKS
Content: 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, and prove that every optimal $1$-planar graph has a $K_6$-minor. Furthermore, we characterize optimal $1$-planar graphs having no $K_7$-minor.