  In this paper we deal with the problem of computing the exact crossing number
of almost planar graphs and the closely related problem of computing the exact
anchored crossing number of a pair of planar graphs. It was shown by [Cabello
and Mohar, 2013] that both problems are NP-hard; although they required an
unbounded number of high-degree vertices (in the first problem) or an unbounded
number of anchors (in the second problem) to prove their result. Somehow
surprisingly, only three vertices of degree greater than 3, or only three
anchors, are sufficient to maintain hardness of these problems, as we prove
here. The new result also improves the previous result on hardness of joint
crossing number on surfaces by [Hlin\v{e}n\'y and Salazar, 2015]. Our result is
best possible in the anchored case since the anchored crossing number of a pair
of planar graphs with two anchors each is trivial, and close to being best
possible in the almost planar case since the crossing number is efficiently
computable for almost planar graphs of maximum degree 3 [Riskin 1996, Cabello
and Mohar 2011].
