Two plane embeddings of one connected planar graph have nonisomorphic dual multigraphs

Embed the three $s$-$t$ paths in cyclic order. Write $f_{12},f_{23},f_{31}$ for the faces between the indicated path pairs. In the dual of The plane dual multigraph, with a vertex for each face and one crossing edge for each primal edge, the path lengths $1,2,3$ give respectively that many parallel dual edges between the two faces on each side of the path. Before accounting for $sw$, the degrees of $f_{12},f_{23},f_{31}$ are $3,5,4$. The bridge may be drawn into any one of these face sectors at $s$; it then gives a dual loop at that face.

In one embedding draw $sw$ into $f_{12}$; the dual degree multiset is then $\{5,5,4\}$. In another draw it into $f_{31}$; the dual degree multiset is $\{3,5,6\}$. These multisets differ, since a loop contributes two to its incident degree. By [F1] the two dual multigraphs are nonisomorphic, although both arise from the same connected planar graph $G$.