  A parallel minor is obtained from a graph by any sequence of edge
contractions and parallel edge deletions. We prove that, for any positive
integer k, every internally 4-connected graph of sufficiently high order
contains a parallel minor isomorphic to a variation of K_{4,k} with a complete
graph on the vertices of degree k, the k-partition triple fan with a complete
graph on the vertices of degree k, the k-spoke double wheel, the k-spoke double
wheel with axle, the (2k+1)-rung Mobius zigzag ladder, the (2k)-rung zigzag
ladder, or K_k. We also find the unavoidable parallel minors of 1-, 2-, and
3-connected graphs.
