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arXiv · 2504.11943

Dividing sums of cycles in the semiring of functional digraphs

Florian Bridoux·Christophe Crespelle·Thi Ha Duong Phan·Adrien Richard
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Abstract

Functional digraphs are unlabelled finite digraphs where each vertex has exactly one out-neighbor. They are isomorphic classes of finite discrete-time dynamical systems. Endowed with the direct sum and product, functional digraphs form a semiring with an interesting multiplicative structure. For instance, we do not know if the following division problem can be solved in polynomial time: given two functional digraphs $A$ and $B$, does $A$ divide $B$? That $A$ divides $B$ means that there exists a functional digraph $X$ such that $AX$ is isomorphic to $B$, and many such $X$ can exist. We can thus ask for the number of solutions $X$. In this paper, we focus on the case where $B$ is a sum of cycles (a disjoint union of cycles, corresponding to the limit behavior of finite discrete-time dynamical systems). There is then a na\"ive sub-exponential algorithm to compute the non-isomorphic solutions $X$, and our main result is an improvement of this algorithm which has the property to be polynomial when $A$ is fixed. It uses a divide-and-conquer technique that should be useful for further developments on the division problem.

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Florian Bridoux, Christophe Crespelle, Thi Ha Duong Phan, Adrien Richard. 2025-04-16. Dividing sums of cycles in the semiring of functional digraphs. https://doi.org/10.1007/s11047-026-10065-w

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