An algorithm for calculating the Nielsen number on surfaces with boundary

Let $f:M\to M$ be a self-map of a hyperbolic surface with boundary. The Nielsen number, $N(f)$, depends only on the induced map $f_{\#}$ of the fundamental group, which can be viewed as a free group on $n$ generators, $a_1,\dotsc,a_n$. 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 $f_{\#}$ is $W$-characteristic. We define an algebraic condition on the $f_{\#}(a_i)$ and show that ``most'' maps satisfy this condition and that all maps which satisfy this condition are $W$-characteristic. If $f_{\#}$ is $W$-characteristic, we present an algorithm for calculating $N(f)$ and prove that the inequality $|L(f)-\chi(M)|\le N(f)-\chi(M)$ holds, where $L(f)$ denotes the Lefschetz number of $f$ and $\chi(M)$ the Euler characteristic of $M$, thus answering in part a question of Jiang and Guo.