# Cartesian products of directed graphs with loops

## TL;DR

This paper proves that connected directed graphs with loops can be uniquely decomposed into prime factors via Cartesian or weak Cartesian products, with an efficient linear-time algorithm for finite graphs.

## Contribution

It establishes a unique factorization theorem for directed graphs with loops and provides a linear-time algorithm for finite cases.

## Key findings

- Unique prime factorization for directed graphs with loops.
- Linear-time algorithm for finite graph factorization.
- Applicability to both finite and infinite graphs.

## Abstract

We show that every nontrivial finite or infinite connected directed graph with loops and at least one vertex without a loop is uniquely representable as a Cartesian or weak Cartesian product of prime graphs. For finite graphs the factorization can be computed in linear time and space.