Grids and their minors

AbstractWe define graphs Yn such that none of these Yn has a minor isomorphic to K8 and for every number k and surface S not all of these graphs are in the completion of k-vertex extensions of graphs embeddable in S. This disproves a corresponding conjecture of N. Robertson and P. D. Symour [in “Progress in Graph Theory” (J. Adian Bondy and U. S. R. Murty, Eds.), pp. 399–406, Academic Press, San Diego/Toronto, 1984]