  A recent result of Bokal et al. [Combinatorica, 2022] proved that the exact
minimum value of c such that c-crossing-critical graphs do not have bounded
maximum degree is c=13. The key to that result is an inductive construction of
a family of 13-crossing-critical graphs with many vertices of arbitrarily high
degrees. While the inductive part of the construction is rather easy, it all
relies on the fact that a certain 17-vertex base graph has the crossing number
13, which was originally verified only by a machine-readable computer proof. We
provide a relatively short self-contained computer-free proof of the latter
fact. Furthermore, we subsequently generalize the critical construction in
order to provide a definitive answer to a remaining open question of this
research area; we prove that for every c>=13 and integers d,q, there exists a
c-crossing-critical graph with more than q vertices of each of the degrees
3,4,...,d.
