{"type":"standard","title":"Dual graph","displaytitle":"Dual graph","namespace":{"id":0,"text":""},"wikibase_item":"Q2294516","titles":{"canonical":"Dual_graph","normalized":"Dual graph","display":"Dual graph"},"pageid":2536864,"thumbnail":{"source":"https://upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Duals_graphs.svg/320px-Duals_graphs.svg.png","width":320,"height":228},"originalimage":{"source":"https://upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Duals_graphs.svg/313px-Duals_graphs.svg.png","width":313,"height":223},"lang":"en","dir":"ltr","revision":"1283677333","tid":"e37b73d7-1020-11f0-9e0d-f01045de1a3e","timestamp":"2025-04-03T00:16:29Z","description":"Graph representing faces of another graph","description_source":"local","content_urls":{"desktop":{"page":"https://en.wikipedia.org/wiki/Dual_graph","revisions":"https://en.wikipedia.org/wiki/Dual_graph?action=history","edit":"https://en.wikipedia.org/wiki/Dual_graph?action=edit","talk":"https://en.wikipedia.org/wiki/Talk:Dual_graph"},"mobile":{"page":"https://en.m.wikipedia.org/wiki/Dual_graph","revisions":"https://en.m.wikipedia.org/wiki/Special:History/Dual_graph","edit":"https://en.m.wikipedia.org/wiki/Dual_graph?action=edit","talk":"https://en.m.wikipedia.org/wiki/Talk:Dual_graph"}},"extract":"In the mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph.","extract_html":"
In the mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph.
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{"fact":"A cat uses its whiskers for measuring distances. The whiskers of a cat are capable of registering very small changes in air pressure.","length":134}
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