a PL structure, i.e. a family of compatible triangulations (see e.g. [@Hu]); by our convention above, all polyhedra are compact. The polyhedron triangulated by a simplicial complex $K$ is denoted $|K|$. All [*maps*]{} between polyhedra shall be assumed piecewise linear, unless stated otherwise. Two embeddings $f$, $g$ of a polyhedron in a sphere are considered [*equivalent*]{} if they are related by an isotopy $h_t$ of the sphere (i.e. $h_0=\operatorname{id}$ and $h_1f=g$).

Graphs
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The complete graph $K_5$ and the complete bipartite graph $K_{3,3}$ (also known as the utilities graph) are shown in Fig. 1. They can be viewed as the $1$-skeleton $(\Delta^4)^{(1)}$ of the $4$-simplex and the join $(\Delta^2)^{(0)}*(\Delta^2)^{(0)}$ of two copies of the three-point set.

A graph $G$ contains no subgraph that is a subdivision of $K_5$ or $K_{3,3}$ iff $|G|$ embeds in $S^2$.

Known proofs of the ‘only if’ part involve exhaustion of cases. A relatively short proof was given by Yu. Makarychev [@Mak] and further developed in [@Sk-A1]. An interesting configuration space approach was suggested by Sarkaria [@Sa3] (but beware of an error, pointed out in [@Sk-M]). Given the considerable difficulty of the result, it is astonishing that besides Kuratowski’s own proof [@Ku] (announced in 1929), there 