On a graph property generalizing planarity and flatness

We introduce a topological graph parameter σ(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 σ(G) ≤ 1, 2, 3, and 4, respectively. Among several other theorems, we show that if H is a minor of G, then σ(H) ≤ σ(G), that σ(Kn) = n−1, and that if H is the suspension of G, then σ(H) = σ(G)+1. Furthermore, we show that μ(G) ≤ σ(G)+2 for each graph G. Here μ(G) is the graph parameter introduced by Colin de Verdière in [2].