Let
(
C
,
0
)
(C,0)
be a reduced curve germ in a normal surface singularity
(
X
,
0
)
(X,0)
. The main goal is to recover the delta invariant
δ
(
C
)
\delta (C)
of the abstract curve
(
C
,
0
)
(C,0)
from the topology of the embedding
(
C
,
0
)
⊂
(
X
,
0
)
(C,0)\subset (X,0)
. We give explicit formulae whenever
(
C
,
0
)
(C,0)
is minimal generic and
(
X
,
0
)
(X,0)
is rational (as a continuation of some previous papers by the authors).
Additionally, in this case, we prove that if
(
X
,
0
)
(X,0)
is a quotient singularity, then
δ
(
C
)
\delta (C)
only admits the values
r
−
1
r-1
or
r
r
, where
r
r
is the number or irreducible components of
(
C
,
0
)
(C,0)
. (
δ
(
C
)
=
r
−
1
\delta (C)=r-1
realizes the extremal lower bound, valid only for ‘ordinary
r
r
–tuples’.)