Minors in large almost-5-connected non-planar graphs

It is shown that every sufficiently large almost-5-connected non-planar graph contains a minor isomorphic to an arbitrarily large graph from one of six families of graphs. The graphs in these families are also almost-5-connected, by which we mean that they are 4-connected and all 4-separations contain a “small” side. As a corollary, every sufficiently large almost-5-connected non-planar graph contains both a K3, 4-minor and a -minor. The connectivity condition cannot be reduced to 4-connectivity, as there are known infinite families of 4-connected non-planar graphs that do not contain a K3, 4-minor. Similarly, there are known infinite families of 4-connected non-planar graphs that do not contain a -minor. © 2012 Wiley Periodicals, Inc. (Contract grant sponsors: C8C Foundation; Inamori Foundation; Kayamori Foundation (K. K.).)