  Stellar halos observed around normal galaxies are extended and faint stellar
structures formed by debris of tidally disrupted dwarf galaxies accreted over
time by the host galaxy. Around dwarf galaxies, these stellar halos may not
exist if all the accreted satellites are dark halos without stars. However, if
a stellar halo is found in sufficiently small mass dwarfs, the whole stellar
halo is composed of tidal debris of fossil galaxies, and we refer to it as
ghostly halo. Fossil galaxies are called so because they formed most of their
stars before the epoch of reionization, and have been identified as the
ultra-faint dwarf galaxies found around the Milky Way and M31.
  In this paper we carry out semi-analytical simulations to characterize the
sizes and stellar masses of ghostly stellar halos in dwarf galaxies as a
function of their dark matter halo mass. By comparing the models to
observations of six isolated dwarf galaxies in the Local Group showing evidence
of extended stellar halos, we are able to constrain the star formation
efficiency in fossil galaxies. We find that at redshift z~6, dark matter halos
in the mass range 10^7-10^9 M_sun have a mean star formation efficiency f_* =
M_*/M_{dm} ~ 0.05%-0.1% nearly constant as a function of the dark matter halo
mass. There is a tentative indication of a sharp increase of f_* at M_{dm} ~
10^6-10^7 M_sun, but it is very uncertain and based on only one dwarf galaxy
(Leo~T).
