  Endowing the set of functional graphs (FGs) with the sum (disjoint union of
graphs) and product (standard direct product on graphs) operations induces on
FGs a structure of a commutative semiring $\ring$. The operations on $\ring$
can be naturally extended to the set of univariate polynomials $\ring[X]$ over
$\ring$. This paper provides a polynomial time algorithm for deciding if
equations of the type $AX=B$ have solutions when $A$ is just a single cycle and
$B$ a set of cycles of identical size. We also prove a similar complexity
result for some variants of the previous equation.
