Let \(M\) be a compact manifold and \(f: M \to M\) a map, then \(MF[f]\) is defined to be the minimum number of fixed points among all maps homotopic to \(f\). For manifolds of dimension at least 3, the number \(MF[f]\) can be calculated, in principal, because it equals the Nielsen number \(N(f)\) of the map \(f\). For maps of surfaces, \(MF[f]\) does not equal \(N(f)\) in general and, in fact, the author [Pac. J. Math. 126, 81-123 (1987; Zbl 0599.55001)] showed that there are maps on the pants surface (disc with two holes) with \(MF[f]\) arbitrarily larger than \(N(f)\). He obtained this result as a consequence of an algorithm for calculating \(MF[f]\) for all maps of the pants surface. In the present paper, he abstracts the ideas from his earlier paper to produce what he describes as “a geometric scheme, but not an algorithm” for computing \(MF[f]\) for a map on any surface with negative Euler characteristic and non-empty boundary. A consequence of the geometric scheme is an embedding theorem that allows the author to embed his pants examples in every surface with negative Euler characteristic and non-empty boundary, to demonstrate that, for these surfaces as well, there are maps for which \(MF[f]\) is arbitrarily larger than \(N(f)\). Recently, Boju Jiang [Commutativity and Wecken properties for fixed points on surfaces and 3-manifolds, Topology Appl. (to appear)] has used entirely different techniques to extend the embedding theorem to all surfaces with negative Euler characteristic, including those with empty boundary.