The plane dual multigraph, with a vertex for each face and one crossing edge for each primal edge

Let $G$ be a connected plane graph with face set $F(G)$ (Plane embeddings of finite simple graphs, their faces, facial boundary walks and lengths (counting a bridge twice), and planar graphs). Its dual multigraph $G^*$ has vertex set $F(G)$ and one edge $e^*$ for every primal edge $e$. If the two local sides of $e$ belong to faces $f$ and $g$, then $e^*$ has endpoints $f,g$.

By Face frontiers are unions of whole edges; a cycle edge borders two faces and a bridge borders one, a bridge has the same face on both local sides, so its dual edge is a loop. Different primal edges may have the same incident face pair, so their duals may be parallel. These are permitted by Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph. The definition is attached to the fixed plane embedding: different embeddings of the same abstract planar graph may have nonisomorphic dual multigraphs.