Let
f
:
M
→
M
f:M\to M
be a self-map of a hyperbolic surface with boundary. The Nielsen number,
N
(
f
)
N(f)
, depends only on the induced map
f
#
f_{\#}
of the fundamental group, which can be viewed as a free group on
n
n
generators,
a
1
,
…
,
a
n
a_1,\dotsc ,a_n
. We determine conditions for fixed points to be in the same fixed point class and if these conditions are enough to determine the fixed point classes, we say that
f
#
f_{\#}
is
W
W
-characteristic. We define an algebraic condition on the
f
#
(
a
i
)
f_{\#}(a_i)
and show that “most” maps satisfy this condition and that all maps which satisfy this condition are
W
W
-characteristic. If
f
#
f_{\#}
is
W
W
-characteristic, we present an algorithm for calculating
N
(
f
)
N(f)
and prove that the inequality
|
L
(
f
)
−
χ
(
M
)
|
≤
N
(
f
)
−
χ
(
M
)
|L(f)-\chi (M)|\le N(f)-\chi (M)
holds, where
L
(
f
)
L(f)
denotes the Lefschetz number of
f
f
and
χ
(
M
)
\chi (M)
the Euler characteristic of
M
M
, thus answering in part a question of Jiang and Guo.