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

TL;DR

This paper surveys criteria and obstructions for approximating graph maps by embeddings in the plane, discussing the van Kampen obstruction's role and extending the concepts to higher dimensions with algorithmic implications.

Contribution

It introduces and discusses the van Kampen obstruction for graph embeddings, including its completeness and higher-dimensional generalizations, with algorithmic applications.

Findings

Criteria for approximability by embeddings presented

Van Kampen obstruction discussed and its completeness established

Higher-dimensional van Kampen obstruction introduced and analyzed

Abstract

A map $\varphi:K\to R^2$ of a graph $K$ is approximable by embeddings, if for each $\varepsilon>0$ there is an $\varepsilon$-close to $\varphi$ 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).