On solving basic equations over the semiring of functional digraphs
Discrete Mathematics
2025-11-26 v8 Combinatorics
Abstract
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.
Cite
@article{arxiv.2402.16923,
title = {On solving basic equations over the semiring of functional digraphs},
author = {Alberto Dennunzio and Enrico Formenti and Luciano Margara and Sara Riva},
journal= {arXiv preprint arXiv:2402.16923},
year = {2025}
}
Comments
Final version accepted by DMTCS; added a linefeed before 'Clearly' in the before last page, as asked by the editor