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 R. The operations on R can be naturally extended to the set of univariate polynomials R[X] over R. 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：赋予函数图集合以加法运算（图的不交并）和乘法运算（图的标准直积运算），可诱导出函数图上的交换半环结构R。R上的运算可自然扩展到R上的一元多项式环R[X]。本文提出了一个多项式时间算法，用于判定当A仅为单个环且B为相同尺寸环的集合时，AX=B型方程是否具有解。我们还证明了前述方程某些变体的类似复杂度结果。