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  "body": "\n## Definition\n\nLet $G$ be a [[planar graph|029JQEK15B]] together with a fixed [[embedding|CG91WNMG2T]] in the plane, called a plane graph.\nThe faces of $G$ are the connected components of the complement of the drawing in the plane.\nThe dual $G^*$ is the [[multigraph|01M5T0KDG7]], possibly with loops, defined as follows.\n\n1. The vertices of $G^*$ are the faces of $G$.\n2. For each edge $e$ of $G$ there is one edge $e^*$ of $G^*$.\n3. The ends of $e^*$ are the faces on the two sides of $e$; if the same face lies on both sides, $e^*$ is a loop.\n\n## Remarks\n\nExample: draw $K_4$ in the plane as a triangle with a central vertex joined to the corners.\nThis drawing has $4$ faces, and every two of them share exactly one edge, so its dual is again $K_4$.\nNon-example: $K_5$ has no dual in this sense, since it is not planar and so it has no plane embedding, which the setting requires.\nAn edge $e$ is a [[bridge|0D81KWH9ZM]] exactly when $e^*$ is a loop.\nIf $G$ is connected, then $(G^*)^*$ is isomorphic to $G$.\nA set of edges is a cycle of $G$ exactly when the dual edges form a minimal cut of $G^*$.\nDifferent embeddings of the same planar graph can have non-isomorphic duals.\n",
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