  Weakly separated collections arise in the cluster algebra derived from the
Pl\"ucker coordinates on the nonnegative Grassmannian. Oh, Postnikov, and
Speyer studied weakly separated collections over a general Grassmann necklace
$\mathcal{I}$ and proved the connectivity of every exchange graph. Oh and
Speyer later introduced a generalization of exchange graphs that we call
$\mathcal{C}$-constant graphs. They characterized these graphs in the smallest
two cases. We prove an isomorphism between exchange graphs and a certain class
of $\mathcal{C}$-constant graphs. We use this to extend Oh and Speyer's
characterization of these graphs to the smallest four cases, and we present a
conjecture on a bound on the maximal order of these graphs. In addition, we
fully characterize certain classes of these graphs in the special cases of
cycles and trees.
