Non-planar extensions of subdivisions of planar graphs
Abstract
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∈ V(G) such that no facial cycle of G contains both u and v, or (ii) G'=G+u1v1+u2v2 for some distinct vertices u1,u2,v1,v2∈ V(G) such that u1,u2,v1,v2 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 X is planar for some set X⊂eq V(G) of size at most k, but H Y is non-planar for every set Y⊂eq V(H) of size at most k.