A graph is planar if it can be drawn on the plane in such a way that its edges do not cross each other.
A graph is $k$-planar if it can be drawn on the plane in such a way that each of its edges is crossed at most $k$ times.
There is also a concept of almost planar graphs that relies on edge deletions or contractions.
I am dealing with another kind of graphs: they can be drawn in the plane in such a way that at most $k$ pairs of edges intersect.
Is there a terminology for such graphs?
Best Answer
These are the graphs with pairwise crossing number or pair-crossing number at most $k$. Note that it is an open problem whether the pair-crossing number is actually equal to the usual crossing number of a graph. See Crossing number, pair-crossing number, and expansion and The Graph Crossing Number and its Variants: A Survey for more info.