  A $c$-crossing-critical graph is one that has crossing number at least $c$
but each of its proper subgraphs has crossing number less than $c$. Recently, a
set of explicit construction rules was identified by Bokal, Oporowski, Richter,
and Salazar to generate all large $2$-crossing-critical graphs (i.e., all apart
from a finite set of small sporadic graphs). They share the property of
containing a generalized Wagner graph $V_{10}$ as a subdivision.
  In this paper, we study these graphs and establish their order, simple
crossing number, edge cover number, clique number, maximum degree, chromatic
number, chromatic index, and treewidth. We also show that the graphs are
linear-time recognizable and that all our proofs lead to efficient algorithms
for the above measures.
