🤖 AI Summary
This paper investigates the solvability of the unary linear equation $AX = B$ over the commutative semiring of functional digraphs (FGs), where $A$ is a unicyclic digraph and $B$ is a disjoint union of cycles of equal length. First, it establishes an algebraic and polynomial-theoretic framework for FGs—marking the inaugural formalization of such structures in this semiring. Second, it devises a polynomial-time decision algorithm based on graph isomorphism testing and cyclic decomposition, with time complexity $O(n^k)$. The algorithm leverages the semiring’s operations—addition as disjoint union and multiplication as standard tensor product—to precisely characterize solvability for unicyclic coefficient matrices. Finally, the approach is extended to multiple equation variants, all remaining polynomial-time solvable. This work provides the first efficient decision paradigm for graph algebraic equations, advancing foundational theory in semiring-based graph algebra.
📝 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.