In general, different embeddings of the same planar graph can yield nonisomorphic dual graphs. A uniquely embeddable graph is a planar graph that has a unique dual graph up to isomorphism regardless of the underlying embedding used to construct it.
Two embeddings of a planar graph are equivalent if there is a homeomorphism of the sphere sending one drawing to the other. A uniquely embeddable graph therefore has all its embeddings on the sphere homeomorphic to one another.
Whitney (1932) proved that polyhedral graphs are uniquely embeddable.
The numbers of uniquely embeddable connected graphs on 
, 2, ... vertices are 1, 1, 2, 6, 15, 51, 206, 1297, 11742,
143095, 2056120, 32337106, ... (OEIS A372853),
the first few of which are illustrated above. The largest numbers of planar embeddings
for graphs on 
,
2, ... vertices are 1, 1, 1, 1, 2, 6, 24, 80, 240, 1080, 3780, 13440, ... (OEIS A372854).
Not all trees are uniquely embeddable; the unique smallest three that is not uniquely embeddable has 7 vertices and is illustrated above. There are four 8-vertex trees that are not uniquely embeddable, including the 4-centipede graph.
Fleischner (1973) found a characterization of uniquely embeddable graphs for labeled graphs. A connected planar
graph 
(involving labeling in some way) is uniquely embeddable in the plane iff 
is one of the following graphs:
1. 
is homeomorphic to a 3-connected graph,
2. 
is homeomorphic to the complete
bipartite graph 
,
3. 
is homeomorphic to the triangle
graph 
,
4. 
is homeomorphic to 
or 
,
5. 
is homeomorphic to 
, or
6. 
is homeomorphic to 
with 
and 
.
