Apex-outerplanar minor conjecture

Let $H$ be an outerplanar graph, and let $H^+$ denote the graph obtained from $H$ by adding an apex vertex adjacent to every vertex of $H$. Apex-outerplanar minor conjecture. Every graph with minimum degree at least $|V(H)|=|V(H^+)|-1$ contains $H^+$ as a minor. The theorem in the paper proves this for a large class of outerplanar graphs, while the full conjecture remains open.