Let 
be the cycle rank of a graph 
, 
be the cocycle rank, and
the relative complement 
of a subgraph 
of 
be defined as that subgraph obtained by deleting the lines
of 
.
Then a graph 
is a combinatorial dual of 
if there is a one-to-one
correspondence between their sets of lines such that for any choice 
and 
of corresponding subsets of lines,
 |
where 
is the subgraph of 
with the line set 
.
Whitney (1932) showed that the geometric dual graph and combinatorial dual graph are equivalent for planar graphs (Fleischner 1973; Harary 1994, p. 115), and so may simply be called "the" dual graph. Also, a graph is planar iff it has a combinatorial dual (Harary 1994, p. 115).
