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

Abstract

A map $\phi :K\to R^2$ of a graph $K$ is approximable by embeddings, if for each $\epsilon >0$ there is an $\epsilon$-close to $\phi$ embedding $f:K\to R^2$. Analogous notions were studied in computer science under the names of cluster planarity and weak simplicity. This short survey is intended not only for specialists in the area, but also for mathematicians from other areas. 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).

11 pages, 6 figures, minor corrections

References in corpus (7)

• Embedding and knotting of manifolds in Euclidean spaces
• Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems
• Hardness of embedding simplicial complexes in ${\mathbb{R}}^d$
• Sphere eversions and realization of mappings
• Eliminating Higher-Multiplicity Intersections, III. Codimension 2
• Toward the Hanani-Tutte Theorem for Clustered Graphs
• Recognizing Weakly Simple Polygons

Cited by in corpus (1)

• Invariants of graph drawings in the plane