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  "body": "\n## Summary\n\nIn graph theory, a **Pfaffian orientation** of an undirected graph assigns a direction to each edge, so that certain cycles (the \"even central cycles\") have an odd number of edges in each direction. When a graph has a Pfaffian orientation, the orientation can be used to count the [[perfect matchings|05ZPH8KF9K]] of the graph. This is the main idea behind the FKT algorithm for counting perfect matchings in [[planar graphs|029JQEK15B]], which always have Pfaffian orientations. More generally, every graph that does not have the utility graph $K_{3,3}$ as a graph minor has a Pfaffian orientation, but $K_{3,3}$ does not, nor do infinitely many other minimal non-Pfaffian graphs.\n",
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