## Abstract

Let *X* be an aspherical polyhedron of the homotopy type
of the figure-eight and let *f* : *X* → *X* be a self-map. The Wagner
algorithm [Trans. Amer. Math. Soc. 351 (1999), 41–62] provides computations for the Nielsen number of
self-maps of *X* satisfying the remnant condition. If *f* is without
remnant, then using the concept of mutant by Jiang [Math. Ann. 311 (1998), 467–479] we may assume
that *f*_{#}(*b*) is an initial segment of *f*_{#}(*a*), where *f*_{#} is
the induced endomorphism of π_{1}(*X*) and *a*, *b* are generators of
π_{1}(*X*). Let *f*_{#}(*b*) = *U* and *f*_{#}(*a*) = *U ^{n}*

*R*, where

*n*is the maximal such positive integer. If

*R*is not an initial segment of

*U*, we say that

*f*is of Type

*Y*. In this paper, we prove that if

*f*is of Type

*Y*, then

*f*can be mutated either to a map that has remnant or to an exceptional form for which we can calculate the Nielsen number directly. Not all self-maps of

*X*are of Type

*Y*. However, making use of the results in this paper, an algorithm is presented by Kim [J. Pure Appl. Algebra 216 (2012), 1652–1666] that does compute the Nielsen number for all self-maps of

*X*.

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