We present and analyse the photometric properties of a nearly complete sample of blue compact dwarf (BCD) and irregular galaxies in the Virgo cluster from multi-band SDSS images. Our study intends to shed light on the ongoing debate of whether a structural evolution from present-day star-forming dwarf galaxies in a cluster environment into ordinary early-type dwarf galaxies is possible based on the structural properties. For this purpose, we decompose the surface brightness profiles of the BCDs into the luminosity contribution of the starburst component and that of their underlying low surface brightness (LSB) host. The latter dominates the stellar mass of the BCD. We find that the LSB-components of the Virgo BCDs are structurally compatible with the more compact half of the Virgo early-type dwarfs, except for a few extreme BCDs. Thus, after termination of starburst activity, the BCDs will presumably fade into galaxies that are structurally similar to ordinary early-type dwarfs. In contrast, the irregulars are more diffuse than the BCDs and are structurally similar to the more diffuse half of the Virgo early-type dwarfs. Therefore, the present-day Virgo irregulars are not simply non-starbursting BCDs. If starbursts in cluster BCDs are transient phenomena with a duration of ~100 Myr or less, during which the galaxies could not travel more than ~100 kpc, then a substantial number of non-starbursting counterparts of these systems must populate the same spatial volume, namely the Virgo cluster outskirts. The majority of them would have to be early-type dwarfs, based on the abundance of different galaxy types with similar colours and structural parameters to the LSB-components of the BCDs. However, most Virgo BCDs have redder LSB-host colours and a less prominent starburst than typical field BCDs, preventing a robust conclusion on possible oscillations between BCDs and early-type dwarfs.
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