  This paper proposes a new, visual method to study numerical semigroups and
the Frobenius problem. The method is based on building a so-called reduction
graph, whose nodes usually correspond to monogenic semigroups, and whose edges
can have multiple inputs and outputs. If such a construction is possible, then
determining whether the studied semigroup is symmetric, or finding explicit
forms of its Ap\'ery set and Hilbert series, is reduced to straightforward
computations assisted by a MAPLE program we made available on arXiv. This
approach applies to many of the cases considered in literature, including
semigroups generated by arithmetic and geometric sequences, compound sequences,
progressions of the form $a^n, a^n + a, \ldots, a^n + a^{n-1}$, triangular and
tetrahedral numbers, certain Fibonacci triplets, etc.
  After explaining the general approach in more detail, the paper studies the
types of edges that can be used as building blocks of a reduction graph, as
well as a series of operations that serve to modify or combine valid reduction
graphs. In the end of the paper, we use these techniques to solve the Frobenius
problem for 7 new classes of numerical semigroups.
