Several differing definitions of almost planar (as well as nearly planar) have been
used in the literature (cf. Lipton et al. 2016).

For example, Gubser (1996) defines an almost planar graph as a graph for which either or is planar, where denotes edge deletion and edge contraction.

Following Karpov (2013), call a -planar graph a graph that can be drawn on the plane such that
any edge intersects at most other edges (Pach and Tóth 1997, Karpov 2013). A 0-planar
graph then corresponds to a planar graph and a 1-planar
graph can be termed an almost planar graph (Karpov 2013), a convention adopted here.

Letting
be the maximum number of edges in a bipartite
almost planar graph on vertices, then

(1)

(Karpov 2013). It follows that the minimum degree of any 1-planar bipartite graph is at most 5 and its vertex count is at least 16. It appears that none
of the 41 quintic bipartite graphs on 16 vertices is 1-planar and the smallest known
example of a quinticbipartite
1-planar graph as of Sep. 2022 has 32 vertices (illustrated below; LeechLattice
2022).

Gubser, B. S. "A Characterization of Almost-Planar Graphs." Combin. Prob. Comput.5, 227-245, 1996.Karpov,
D. V. "Upper Bound on the Number of Edges of an Almost Planar Bipartite
Graph." 3 Jul 2013. https://arxiv.org/abs/1307.1013.LeechLattice.
"How to Construct a 5-Regular 1-Planar Bipartite Graph?" Sep. 10,
2022. https://mathoverflow.net/questions/430153/how-to-construct-a-5-regular-1-planar-bipartite-graph.Lipton,
M.; Mackall, E; Mattman, T. W.; Pierce, M.; Robinson, S.; Thomas, J.; and Weinschelbaum,
I. "Six Variations on a Theme: Almost Planar Graphs." 5 Aug 2016. https://arxiv.org/abs/1608.01973.Pach,
J. and Tóth. "Graphs Drawn With Few Crossing Per Edge." Combinatorica17,
427-439, 1997.