Stability of intersections of graphs in the plane and the van Kampen obstruction

A map φ : K → R2 of a graph K is approximable by embeddings, if for each ε > 0 there is an ε-close to φ embedding f : K → R2. Analogous notions were studied in computer science under the names of cluster planarity and weak simplicity. In this survey we present criteria for approximability by embeddings (P. Minc, 1997, M. Skopenkov, 2003) and their algorithmic corollaries. We introduce the van Kampen (or Hanani-Tutte) obstruction for approximability by embeddings and discuss its completeness. We discuss analogous problems of moving graphs in the plane apart (cf. S. Spież and H. Toruńczyk, 1991) and finding closest embeddings (H. Edelsbrunner). We present higher dimensional van Kampen obstruction, its completeness result and algorithmic corollary (D. Repovš and A. Skopenkov, 1998).