  This paper aims to contribute to validate, for numerical semigroups of
reasonably large genus, the so-called Conjecture of Wilf. There is no
counter-example for the conjecture among the over 3*10^{10} numerical
semigroups of genus up to 60, as it has been computationally verified by
Fromentin and Hivert. The computations use the idea of parsing a semigroups
tree, making tests in each node. As a mean to combine parsing of the semigroups
tree with known theoretical results, we introduce the concept of cutting
semigroup. Assume that there exists a property on numerical semigroups that
implies that a semigroup satisfying it necessarily satisfies Wilf's conjecture.
Assume also that, besides, this property is hereditary, that is, if a semigroup
satisfies it, then all its descendants in the semigroups tree also have the
same property. Such properties exist. Interesting ones can be easily deduced
from some deep results of Eliahou. A semigroup satisfying such a property is
called a cutting semigroup (for that property). When looking for
counter-examples to Wilf's conjecture, if a cutting semigroup is found, then it
can be cut off, since no counter example lies among its descendants. One can
therefore consider the trimmed tree obtained by cutting off all the cutting
semigroups. Depending on the properties, these ideas allow in some cases to
avoid parsing a very significant part of the tree and can be applied to other
related problems. For instance, they can be used to find all numerical
semigroups with negative Eliahou number up to a genus bigger than 60, which is
the current record. Computations by Fromentin had shown that there are exactly
5 numerical semigroups in these circumstances.
