K 7-minors in optimal 1-embedded graphs on the projective plane

Every simple optimal 1-embedded graph on a closed surface has a K-t-minor for t <= 6 as Mader proved that every simple graph with n vertices and at least 4n - 9 edges has a K-6-minor. In this paper, we characterize simple optimal 1-embedded graphs on the projective plane that have no K-7-minor. It follows from the aforementioned result that every simple optimal 1-embedded graph on the projective plane which is 5-connected or whose quadrangular subgraph is 4-representative has a K-7-minor. (c) 2024 Elsevier B.V. All rights reserved.