$\partial$-invariant path generators for digraphs

TL;DR

This paper investigates the structure of $oundary$-invariant 3-paths in directed graphs, providing a basis construction, an explicit algorithm, and complexity analysis for understanding their space.

Contribution

It introduces a basis for $oundary$-invariant 3-paths in digraphs and offers an explicit construction and an $O(|V(G)|^5)$ algorithm to compute their dimension and basis.

Findings

Basis of $oundary$-invariant 3-paths consists of trapezohedral paths and their merges.

Explicit construction of the basis is provided.

Algorithm for computing the basis has polynomial time complexity.

Abstract

We study the structure of the space $\Omega_3(G)$ of $\partial$-invariant 3-paths in a directed graph $G$. We prove that $\Omega_3(G)$ admits a basis consisting of trapezohedral paths $\tau_m$ ($m \ge 2$) and their merging images. Moreover, we provide an explicit construction of such a basis and, as a consequence, obtain an algorithm with time complexity $O(|V(G)|^5)$ for computing the dimension and a basis of $\Omega_3(G)$ for any finite digraph.