A biplanar graph is defined as a graph that is the graph union of two planar edge-induced subgraphs. In other words, biplanar graphs are graphs with graph thickness 1 or 2 (e.g., Beineke 1997). Note that by this definition, planar graphs are considered (trivial) biplanar graphs.
Examples of nontrivial biplanar graphs include the Möbius ladders 
,
complete graphs 
, 
, 
, and 
(e.g., Hearon 2016, p. 20) and the complete
bipartite graphs 
,
, 
, and 
. In particualr, the smallest complete graph that is
not biplanar is 
,
and the smallest complete bipartite graphs that are not biplanar are 
, 
, and 
(Hearon 2016, p. 19).
Determining if a graph is biplanar is an NP-complete problem (Mansfeld 1983, Beineke 1997). The precomputed Boolean status for a graph being nontrivially biplanar (i.e., biplanar but not planar) can be obtained for many small named or indexed graphs in the Wolfram Language using GraphData[graph, "Biplanar"].
The vertex count 
, edge count 
, and girth 
of a biplanar graph satisfy
 |
(1)
|
 |
(2)
|
and for a bipartite biplanar graph, satisfy
 |
(3)
|
(Beineke 1997).
