Задачи для исследования о графах на плоскости: устойчивость самопересечений и топологическая гипотеза Тверберга

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. Spiez and H. Torunczyk, 1991) and finding closest embeddings (H. Edelsbrunner). We present higher dimensional van Kampen obstruction, its completeness result and algorithmic corollary (D. Repovs and A. Skopenkov, 1998).
In the second part of this talk I will describe the ‘van Kampen obstruction’ approach to the topological Tverberg probem for the plane.