Minors in optimal 1-embedded graphs

It is known that any optimal 1-planar graph has a K 6 -minor, while those having a K 7 -minor have been completely characterized. Furthermore, one of the authors showed that for any graph H there exists an optimal 1-planar graph having H as a topological minor. In this paper, we study optimal 1-embedded graphs on general closed surfaces and investigate the minors contained in them. In particular, we prove that for any graph H , every optimal 1-embedded graph on a closed surface F 2 whose quadrangular subgraph has sufficiently large representativity has H as a minor. Note that H in our result is not necessarily 1-embeddable on F 2 .