  Properties of the `$k$-equivalent' graph families constructed in Cai,
F\"{u}rer and Immerman, and Evdokimov and Ponomarenko are analysed relative the
the recursive $k$-dim WL method. An extension to the recursive $k$-dim WL
method is presented that is shown to efficiently characterise all such types of
`counterexample' graphs, under certain assumptions. These assumptions are shown
to hold in all known cases.
