  We provide new constructions of Meyniel extremal graphs, which are families
of graphs with the conjectured largest asymptotic cop number. Using spanning
subgraphs, we prove that there are an exponential number of new Meyniel
extremal families with specified degrees. Using a linear programming problem on
hypergraphs, we explore the degrees in families that are not Meyniel extremal.
We give the best-known upper bound on the cop number of vertex-transitive
graphs with a prescribed degree. We find new Meyniel extremal families of
regular graphs with large chromatic number, large diameter, and explore the
connection between Meyniel extremal graphs and bipartite graphs.
