Non-planar extensions of subdivisions of planar graphs

Almost $4$-connectivity is a weakening of $4$-connectivity which allows for vertices of degree three. In this paper we prove the following theorem. Let $G$ be an almost $4$-connected triangle-free planar graph, and let $H$ be an almost $4$-connected non-planar graph such that $H$ has a subgraph isomorphic to a subdivision of $G$. Then there exists a graph $G'$ such that $G'$ is isomorphic to a minor of $H$, and either (i) $G'=G+uv$ for some vertices $u,v\in V(G)$ such that no facial cycle of $G$ contains both $u$ and $v$, or (ii) $G'=G+u_1v_1+u_2v_2$ for some distinct vertices $u_1,u_2,v_1,v_2\in V(G)$ such that $u_1,u_2,v_1,v_2$ appear on some facial cycle of $G$ in the order listed. This is a lemma to be used in other papers. In fact, we prove a more general theorem, where we relax the connectivity assumptions, do not assume that $G$ is planar, and consider subdivisions rather than minors. Instead of face boundaries we work with a collection of cycles that cover every edge twice and have pairwise connected intersection. Finally, we prove a version of this result that applies when $G\backslash X$ is planar for some set $X\subseteq V(G)$ of size at most $k$, but $H\backslash Y$ is non-planar for every set $Y\subseteq V(H)$ of size at most $k$.