were independent contemporary proofs: by L. S. Pontryagin (unpublished[^2], but acknowledged in Kuratowski’s original paper [@Ku]), and by O. Frink and P. A. Smith, announced in 1930 [@FS], [@Wh].

A useful reformulation of Kuratowski’s theorem was suggested by K. Wagner [@Wa]. The following non-standard definition is equivalent to the standard one (see Proposition \[nevo\] below). We call a graph $H$ a [*minor*]{} of a graph $G$, if $H$ is obtained from a subgraph $F$ of $G$ by a sequence of edge contractions, where an edge contraction as a simplicial map $f$ that shrinks one edge $\{v_1,v_2\}$ onto a vertex, provided that $\operatorname{lk}\{v_1\}\cap\operatorname{lk}\{v_2\}=\emptyset$. The latter condition is equivalent to saying that all point-inverses of $|f|$ are points, except for one point inverse, which is an edge.

It is easy to see that $S^2$ modulo an arc is homeomorphic to $S^2$ (cf. Lemma \[trivial\]); hence if $|G|$ embeds in $S^2$ and $H$ is a minor of $G$, then $|H|$ embeds in $S^2$. This observation along with Kuratowski’s theorem immediately imply

A graph $G$ has no minor isomorphic to $K_5$ or $K_{3,3}$ iff $|G|$ embeds in $S^2$.

We are now ready for a more substantial application of minors. We call an embedding $g$ 