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

A map $\varphi:K\to \mathbb R^2$ of a graph $K$ is approximable by embeddings, if for each $\varepsilon>0$ there is an $\varepsilon$-close to $\varphi$ embedding $f:K\to\mathbb R^2$. Analogous notions were studied under the names of cluster planarity and weak simplicity. We present criteria for approximability by embeddings (P. Minc, 1997, M. Skopenkov, 2003) and their algorithmic corollaries. A map $\varphi:K\sqcup L\to \mathbb R^2$ of the disjoint union of graphs $K$ and $L$ is approximable by maps with disjoint images, if for each $\varepsilon>0$ there is an $\varepsilon$-close to $\varphi$ map $f:K\sqcup L\to\mathbb R^2$ such that $f(K)\cap f(L)=\emptyset$. We present open problems on this notion. We introduce the van Kampen (or Hanani-Tutte) obstructions for these properties, as well as their generalizations to higher dimensions and to $r$-tuple intersections. We present the completeness result of this obstruction (D. Repovš and A. Skopenkov, 1998) and its algorithmic corollary.