$\mathrm{\partial }$-invariant path generators for digraphs

We study the structure of the space ${\mathrm{\Omega }}_3(G)$ of $\mathrm{\partial }$-invariant 3-paths in a directed graph $G$. We prove that ${\mathrm{\Omega }}_3(G)$ 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 $O(\mathrm{\mid }V(G){\mathrm{\mid }}^5)$ for computing the dimension and a basis of ${\mathrm{\Omega }}_3(G)$ for any finite digraph.