On graph classes with minor-universal elements

A graph U is universal for a graph class $C\ni U$, if every $G\in C$ is a minor of U. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable into a surface, and graphs forbidding $K_5$, or $K_{3,3}$, or $K_{\infty }$ as a minor. We prove the existence of uncountably many minor-closed classes of countable graphs that do not have a universal element.

Some of our results and questions may be of interest from the finite graph perspective. In particular, one of our side-results is that every $K_5$-minor-free graph is a minor of a $K_5$-minor-free graph of maximum degree 22.