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.

翻译：将函数图（FGs）的集合赋予和（图的互不相交并）与积（图的标准直积）运算，可在FGs上诱导出一个交换半环结构$\ring$。$\ring$上的运算可自然扩展至$\ring$上的一元多项式环$\ring[X]$。本文给出了一个多项式时间算法，用于判定当$A$为单个环、$B$为若干相同大小的环的集合时，形如$AX=B$的方程是否有解。我们还证明了上述方程若干变体具有类似的计算复杂性结论。