Kuratowski's theoremKuratowskisTheorem2001-11-12 13:33:212004-03-27 14:34:55Theoremplanar\usepackage{amssymb}
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\makeatotherA finite graph is planar if and only if it contains no subgraph that is isomorphic to or is a subdivision of $K_5$ or $K_{3,3}$, where $K_5$ is the complete graph of order 5 and $K_{3,3}$ is the complete bipartite graph with 3 vertices in each of the halfs. Wagner's theorem is an equivalent later result.
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Kazimierz Kuratowski.
\newblock Sur le probl{\`e}me des courbes gauches en topologie.
\newblock {\em Fund. Math.}, 15:271--283, 1930.
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%@ARTICLE{cite:kuratowski_planarity,
% author = {Kazimierz Kuratowski},
% title = "Sur le Probl{\`e}me des Courbes Gauches en Topologie",
% journal = {Fund. Math.},
% volume = 15,
% pages = {271--283},
% year = 1930
%}