Kuratowski's and Wagner's theorems

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https://en.wikipedia.org/wiki/Planar_graph

The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of forbidden graphs, now known as Kuratowski's theorem:

A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K₅ (the complete graph on five vertices) or K_3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three, also known as the utility graph).

A subdivision of a graph results from inserting vertices into edges (for example, changing an edge •——• to •—•—•) zero or more times.

 

An example of a graph which doesn't have K₅ or K_3,3 as its subgraph. However, it has a subgraph that is homeomorphic to K_3,3 and is therefore not planar.

Instead of considering subdivisions, Wagner's theorem deals with minors:

A finite graph is planar if and only if it does not have K₅ or K_3,3 as a minor.

 

An animation showing that the Petersen graph contains a minor isomorphic to the K_3,3 graph

Klaus Wagner asked more generally whether any minor-closed class of graphs is determined by a finite set of "forbidden minors". This is now the Robertson–Seymour theorem, proved in a long series of papers. In the language of this theorem, K₅ and K_3,3 are the forbidden minors for the class of finite planar graphs.

https://zh.wikipedia.org/wiki/平面图_(图论)

波兰数学家卡齐米日·库拉托夫斯基提出的一类禁忌准则（指满足某种条件的图就一定无法具有某个性质）中，也包括了平面图的情况。他提出的一个定理说明：

一个有限图（顶点数和边数有限的图）是平面图当且仅当它并不包含一个是{displaystyle K_{5}}（有五个顶点的完全图）或{displaystyle K_{3,3}}（三个顶点的二部图）的分割的子图^[1]。

其中，一个图A是另一个图B的分割是指：A是在B的基础上，在某些边的中间加上顶点：

而得到的新的图。

用图的同胚理论来说，就是：一个有限图是平面图当且仅当这个图不包含任何同胚于或的子图。

这个定理的一般化是罗伯森－西摩定理。