Graphs and obstructions in four dimensions

For any graph G=(V,E) without loops, let C2(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 C2(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 K7 and K3,3,1,1 by a series of ΔY- and YΔ-transformations, C2(G) cannot be embedded into 4-space.