Vertex suppression in 3-connected graphs

To suppress a vertex v in a finite graph G means to delete it and add an edge from a to b if a, b are distinct nonadjacent vertices which formed the neighborhood of v. Let G - -x be the graph obtained from G - x by suppressing vertices of degree at most 2 as long as it is possible; this is proven to be well defined. Our main result states that every 3-connected graph G has a vertex x such that G - -x is 3-connected unless G is isomorphic to K-3,(3), K-2 x K-3, or to a wheel K-1 * C-l for some l >= 3. This leads to a generator theorem for 3-connected graphs in terms of series parallel extensions. (c) 2007 Wiley Periodicals, Inc.