  Structural and dynamical parameters of faint open clusters are probed with
quality 2MASS-photometry and analytical procedures developed for bright
clusters. We derive fundamental parameters of the faint open clusters Lynga2,
BH63, Lyng12 and King20, the last three of which have no prior determinations.
We also focus on the structure and dynamical state of these clusters. 2MASS
photometry with errors smaller than 0.2mag are used to build CMDs, RDPs, CMDs,
luminosity and mass functions. Colour-magnitude filters are used to isolate
probable member stars. Field-star decontamination is applied to Lynga2, Lynga12
and King20. Reddening values are in the range 0.22-1.9, with BH63 the most
reddened object. Ages of Lynga2, King20, Lynga12 and BH63 are ~90, ~200, ~560
and 700Myr, respectively. The RDPs of Lynga12 and King20 are well-represented
by King profiles. Total stellar masses (extrapolating the MFs to stars with
0.08Mo) range from ~340Mo (BH63) to ~2300Mo (Lynga12). Observed masses are
\~1/4$ of these values. In all clusters the core mass function is flatter than
the halo's. Faint open clusters can be probed with 2MASS when associated with
colour-magnitude filters and field-star decontamination. BH63 appears to be in
an advanced dynamical state, both in the core and halo. To a lesser degree the
same applies to King20. Marginal evidence of dynamical evolution is present in
the cores of Lynga2 and Lynga12.
