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"].

Beineke, L. W. "'Biplanar Graphs: A Survey." Computers Math. Appl.34, 1-8, 1997.Hearon, S. M.
"Planar Graphs, Biplanar Graphs and Graph Thickness." Master of Arts thesis.
San Bernadino, CA: California State University, San Bernadino, 2016.Mansfeld,
A. "Determining the Thickness of a Graph is NP-Hard." Math. Proc. Cambridge
Philos. Soc.93, 9-23, 1983.