  We analyze deep CCD images of nearby Blue Compact Dwarf (BCD) galaxies in an
attempt to understand the nature of the progenitors which are hosting the
current burst of star formation. In particular, we ask whether BCDs are hosted
by normal or low-surface-brightness dI galaxies. We conclude that BCDs are in
fact hosted by gas-rich galaxies which populate the extreme
high-central-mass-density end of the dwarf galaxy distribution. Such galaxies
are predisposed to having numerous strong bursts of star formation in their
central regions. In this picture, BCDs can only occur in the minority of dwarf
galaxies, rather than being a common phase experienced by all gas-rich dwarfs.
