  Context. The details of the C/O core structure in white dwarf stars has
mostly remained inaccessible to the technique of asteroseismology, despite
several attempts carried out in the past.
  Aims. We re-assess the potential of asteroseismology for probing the chemical
stratification in white dwarf cores, in light of new highly efficient tools
recently developed for that purpose.
  Methods. Using the forward modeling approach and a new parameterization for
the core chemical stratification in ZZ Ceti stars, we test several situations
typical of the usually limited constraints available, such as small numbers of
observed independent modes, to carry out asteroseismology of these stars.
  Results. We find that, even with a limited number of modes, the core chemical
stratification (in particular, the location of the steep chemical transitions
expected in the oxygen profile) can be determined quite precisely due to the
significant sensitivity of some confined modes to partial reflexion (trapping)
effects. These effects are similar to the well known trapping induced by the
shallower chemical transitions at the edge of the core and at the bottom of the
H-rich envelope. We also find that success to unravel the core structure
depends on the information content of the available seismic data. In some
cases, it may not be possible to isolate a unique, well-defined seismic
solution and the problem remains degenerate.
  Conclusions. Our results establish that constraining the core chemical
stratification in white dwarf stars based solely on asteroseismology is
possible, an opportunity that we have started to exploit.
