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（FGs）加法（图的不交并）和乘法（图的标准直积）运算，使得函数图构成一个交换半环 R。R 上的运算可以自然地扩展到 R 上的单变量多项式集合 R[X]。本文提出了一种多项式时间算法，用于判定当 A 仅为单个环且 B 为一组相同大小的环时，形如 AX=B 的方程是否有解。我们还证明了前述方程某些变体的类似复杂度结果。