Functional graphs (FGs) model the graph structures used to analyze the behavior of functions from a discrete set to itself. In turn, such functions are used to study real complex phenomena evolving in time. As the systems involved can be quite large, it is interesting to decompose and factorize them into several subgraphs acting together. Polynomial equations over functional graphs provide a formal way to represent this decomposition and factorization mechanism, and solving them validates or invalidates hypotheses on their decomposability. The current solution method breaks down a single equation into a series of \emph{basic} equations of the form $A\times X=B$ (with $A$, $X$, and $B$ being FGs) to identify the possible solutions. However, it is able to consider just FGs made of cycles only. This work proposes an algorithm for solving these basic equations for general connected FGs. By exploiting a connection with the cancellation problem, we prove that the upper bound to the number of solutions is closely related to the size of the cycle in the coefficient $A$ of the equation. The cancellation problem is also involved in the main algorithms provided by the paper. We introduce a polynomial-time semi-decision algorithm able to provide constraints that a potential solution will have to satisfy if it exists. Then, exploiting the ideas introduced in the first algorithm, we introduce a second exponential-time algorithm capable of finding all solutions by integrating several `hacks' that try to keep the exponential as tight as possible.

翻译：函数图（FGs）是对从离散集合到其自身的函数行为进行分析时所使用的图结构模型，而这类函数常用于研究随时间演化的真实复杂现象。由于涉及的系统可能规模庞大，将其分解为若干协同作用的子图具有重要意义。基于函数图的多项式方程为这种分解与因式分解机制提供了形式化表达，求解这些方程可验证或否定关于其可分解性的假设。当前求解方法将单一方程拆解为一系列形如$A\times X=B$（其中$A$、$X$和$B$均为函数图）的"基本方程"，以识别可能的解集。然而该方法仅能处理由纯环构成的函数图。本文提出一种针对一般连通函数图求解基本方程的算法。通过利用与消去问题的关联，我们证明解的数量上界与方程系数$A$中环的大小密切相关。消去问题同样贯穿于本文提供的主要算法中。首先提出一个多项式时间半决策算法，该算法能为潜在解（若存在）必须满足的约束条件提供规范表达；继而基于首个算法引入的思想，通过整合若干试图将指数增长最小化的"优化策略"，提出第二个指数时间算法，可完整找出所有解。