  We improve on a recently constructed graphical representation of the
supergravity 7-brane solution and apply this refined representation to re-study
the open string description of the A-D-E-singularities in F-theory on K3. A
noteworthy feature of the graphical representation is that it provides the
complete global branch cut structure of the 7-brane solution which plays an
important role in our analysis. We first identify those groups of branes which
when made to coincide lead to the A-D-E-gauge groups. We next show that there
is always a sufficient number of open BPS strings to account for all the
generators of the gauge group. However, as we will show, there is in general no
one-to-one relation between BPS strings and gauge group generators.
  For the D_{n+4}- and E-singularities, in order to relate BPS strings with
gauge group generators, we make an SU(n+4), respectively SU(5) subgroup of the
D_{n+4}- and E-gauge groups manifest. We find that only for the D-series (and
for the standard A-series) this is sufficient to identify, in a one-to-one
manner, which BPS strings correspond to which gauge group generators.
