Аннотация:
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.
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