  We argue that the ``vortex-finding'' property of maximal center gauge, i.e.
the ability of this gauge to locate center vortices inserted by hand on any
given lattice, is the key to its success in extracting the vortex content of
thermalized lattice configurations. We explain how this property comes about,
and why it is expected not only in maximal center gauge, but also in an
infinite class of gauge conditions based on adjoint-representation link
variables. In principle, the vortex-finding property can be foiled by Gribov
copies. This fact is relevant to a gauge-fixing procedure devised by Kovacs and
Tomboulis, where we show that the loss of center dominance, found in their
procedure, is explained by a corresponding loss of the vortex-finding property.
The association of center dominance with the vortex-finding property is
demonstrated numerically in a number of other gauges.
