  The fraction of star-forming to quiescent dwarf galaxies varies from almost
infinity in the field to zero in the centers of rich galaxy clusters. What is
causing this pronounced morphology-density relation? What do quiescent dwarf
galaxies look like when studied in detail, and what conclusions can be drawn
about their formation mechanism? Here we study a nearly magnitude-complete
sample (-19 < M_r < -16 mag) of 121 Virgo cluster early types with deep
near-infrared images from the SMAKCED project. We fit two-dimensional models
with optional inner and outer components, as well as bar and lens components
(in ~15% of the galaxies), to the galaxy images. While a single S\'ersic
function may approximate the overall galaxy structure, it does not entirely
capture the light distribution of two-thirds of our galaxies, for which
multi-component models provide a better fit. This fraction of complex galaxies
shows a strong dependence on luminosity, being larger for brighter objects. We
analyze the global and component-specific photometric scaling relations of
early-type dwarf galaxies and discuss similarities with bright early and late
types. The dwarfs' global galaxy parameters show scaling relations that are
similar to those of bright disk galaxies. The inner components are mostly
fitted with S\'ersic n values close to 1. At a given magnitude they are
systematically larger than the bulges of spirals, suggesting that they are not
ordinary bulges. We argue that the multi-component structures in early-type
dwarfs are mostly a phenomenon inherent to the disks, and may indeed stem from
environmental processing.
