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

Understand

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.