Context : Structural and dynamical parameters of faint open clusters are probed with quality 2MASS-photometry and analytical procedures developed for bright clusters .

Aims : We derive fundamental parameters of the faint open clusters Lyngå 2 , BH 63 , Lyngå 12 and King 20 , the last three of which have no prior determinations .
We also focus on the structure and dynamical state of these clusters .

Methods : J , H and K _ { s } 2MASS photometry with errors smaller than 0.2 mag are used to build CMDs , radial density profiles , colour-colour diagrams , luminosity and mass functions .
Colour-magnitude filters are used to isolate probable member stars .
Field-star decontamination is applied to Lyngå 2 , Lyngå 12 and King 20 .

Results : Reddening values are in the range 0.22 \leq \mbox { $E ( B - V ) $ } \leq 1.9 , with BH 63 the most reddened object .
Ages of Lyngå 2 , King 20 , Lyngå 12 and BH 63 are \approx 90 , \approx 200 , \approx 560 and \approx 700 Myr , respectively .
The radial density distributions of Lyngå 12 and King 20 are well-represented by King profiles .
Lyngå 2 and BH 63 are very small with core and limiting radii of \approx 0.12 pc and \approx 1.5 pc .
Yet , they fit in the small-radii tail of the open cluster size distribution .
Lyngå 12 and King 20 have \mbox { $R _ { core } $ } \approx 0.43 pc and \mbox { $R _ { lim } $ } \approx 3.9 pc .
Lyngå 2 and Lyngå 12 are inside the Solar circle .
Total stellar masses ( extrapolating the MFs to stars with 0.08 M _ { \odot } ) range from \approx 340 M _ { \odot } ( BH 63 ) to \approx 2300 M _ { \odot } ( Lyngå 12 ) .
Observed masses are \sim 1 / 4 of these values .
In all clusters the core mass function is flatter than the halo ’ s .

Conclusions : Faint open clusters can be probed with 2MASS when associated with colour-magnitude filters and field-star decontamination .
BH 63 appears to be in an advanced dynamical state , both in the core and halo .
To a lesser degree the same applies to King 20 .
Marginal evidence of dynamical evolution is present in the cores of Lyngå 2 and Lyngå 12 .