On a graph property generalizing planarity and flatness

Abstract

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(K n )=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 Verdiere in [2].