Wagner's theorem
In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither K5 (the complete graph on five vertices) nor K3,3 (the utility graph, a complete bipartite graph on six vertices). This was one of the earliest results in the theory of graph minors and can be seen as a forerunner of the Robertson–Seymour theorem.
A planar embedding of a given graph is a drawing of the graph in the Euclidean plane, with points for its vertices and curves for its edges, in such a way that the only intersections between pairs of edges are at a common endpoint of the two edges. A minor of a given graph is another graph formed by deleting vertices, deleting edges, and contracting edges. When an edge is contracted, its two endpoints are merged to form a single vertex. In some versions of graph minor theory the graph resulting from a contraction is simplified by removing self-loops and multiple adjacencies, while in other version multigraphs are allowed, but this variation makes no difference to Wagner's theorem. Wagner's theorem states that every graph has either a planar embedding, or a minor of one of two types, the complete graph K5 or the complete bipartite graph K3,3. (It is also possible for a single graph to have both types of minor.)
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