Graphs and obstructions in four dimensions

For any graph G = ( V , E ) without loops, let C 2 ( G ) denote the regular CW-complex obtained from G by attaching to each circuit C of G a disc. We show that if G is the suspension of a flat graph, then C 2 ( G ) has an embedding into 4-space. Furthermore, we show that for any graph G in the collection of graphs that can be obtained from K 7 and K 3 , 3 , 1 , 1 by a series of Δ Y - and Y Δ -transformations, C 2 ( G ) cannot be embedded into 4-space.