On a graph property generalizing planarity and flatness

We introduce a topological graph parameter s(G), defined for any graph G. This parameter characterizes subgraphs of paths, outerplanar graphs, planar graphs, and graphs that have a flat embedding as those graphs G with s(G)=1,2,3, and 4, respectively. Among several other theorems, we show that if H is a minor of G, then s(H)=s(G), that s(K n )=n-1, and that if H is the suspension of G, then s(H)=s(G)+1. Furthermore, we show that µ(G)=s(G) + 2 for each graph G. Here µ(G) is the graph parameter introduced by Colin de Verdière in [2].