Prove that every dual graph of a planar graph is planar
By David Jones •
It seems obvious, but how to prove it properly? I tried Kuratowski, but got stuck at $K_{3,3}$
$\endgroup$ 51 Answer
$\begingroup$The dual graph of a planar graph is that connecting two regions iff they share a common edge. You can put a "dual-dot" somewhere in the interior of each face. Then you can connect two dual dots for faces that meet along an edge by drawing an arc connecting them which lies inside the two faces of the planar graph in which the dual dots lie, crossing the common edge. If this is done carefully, none of the added arcs will cross each other.
NOTE I had previously confused the line graph with the dual graph, and thanks to @EuYu for pointing that out. I hope this is OK now, as it at least is about the dual graph.
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