Given a planar graph 
with a particular planar embedding,
a geometric dual graph and combinatorial
dual graph can be defined. Whitney (1932) showed that these are equivalent for
planar graphs (Fleischner 1973, Harary 1994. p. 115)
so that one may speak of "the" dual graph 
. The illustration above shows the process of constructing
a geometric dual graph from a planar nonpolyhedral
graph, leading to a multigraph.
While some nonpolyhedral planar graphs have a unique dual, a general planar graph has multiple dual graphs depending on the choice of planar
embedding. A planar graph has a unique embedding (and hence is said to be unqiuely embeddable), and consequently a unique
dual, iff it is a graph
subdivision of a polyhedral graph. The complete bipartite graph 
is an example of a planar nonpolyhedral graph whose
embeddings are all isomorphic, meaning its dual graphs are also isomorphic and it
is uniquely embeddable.
On the other hand, polyhedral graphs have unique dual graphs. The dual graph 
of a polyhedral graph 
has graph vertices each of
which corresponds to a face of 
and each of whose faces corresponds to a graph
vertex of 
.
Two nodes in 
are connected by an graph edge if the corresponding
faces in 
have a boundary graph edge in common. As a result,
each edge of a graph 
has a corresponding dual edge 
in 
corresponding to the edge that connects the two faces on either side of 
, meaning the edge counts are
the same. Combined with the switching of the roles of faces and vertices, this gives
the relations
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
between dual and primal edge, face, and vertex counts. They also of course satisfy the polyhedral formula
 |  |  |
(4)
|
 |  |  |
(5)
|
The dual graph of a wheel graph is itself a wheel (Skiena 1990, p. 147). In general, a graph that is dual to itself is called a self-dual graph.
The notation of duality can be generalized to embeddings other than those in the plane and hence to nonplanar graphs. This is closely related to the concept of double covers.
Graph duals of named graphs are implemented in the Wolfram Language as GraphData[graph, "DualGraph"].
The Tutte polynomial of the dual graph 
of a graph 
is given by
 |
(6)
|
i.e., by swapping the variables of the Tutte polynomial of the original graph.
