Minors in Large Almost-5-Connected Non-Planar Graphs Ken-Ichi Kawarabayashi 1 and John Maharry 2 1 NATIONAL INSTITUTE OF INFORMATICS, 2-1-2, HITOTSUBASHI CHIYODA-KU, TOKYO 101-8430, JAPAN E-mail: k keniti@nii.ac.jp 2 DEPARTMENT OF MATHEMATICS, THE OHIO STATE UNIVERSITY COLUMBUS, OHIO E-mail: maharry@math.ohio-state.edu Received June 10, 2008; Revised July 2, 2011 Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jgt.20637 Abstract: 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 K 3,4 -minor and a K - 6 -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 K 3,4 -minor. Similarly, there are known infinite families of 4-connected non-planar graphs that do not contain a K - 6 -minor.  2011 Wiley Periodicals, Inc. J Graph Theory Contract grant sponsors: C&C Foundation; Inamori Foundation; Kayamori Foundation (K. K.). Journal of Graph Theory  2011 Wiley Periodicals, Inc. 1