Minors of quasi 4-connected graphs

AbstractA minimal point disconnecting set S of a graph G is a nontrivial m-separator, where m=|S|, if the connected components of G—S can be partitioned into two sets each of which has at least two points. A 3-connected graph is quasi 4-connected if it has no nontrivial 3-separators. Let G be a quasi 4-connected graph. It is proved that (i) if G is nonplanar with 7 or more points then G is either isomorphic to C8(1,4) or K5 is a minor of G and (ii) if G is planar with 9 or more points then G is K2 x C5 or K2,2,2 is a minor of G. Some well-known results follow from this as corollaries