$\partial$-invariant path generators for digraphs

TL;DR

It is proved that $\Omega_3(G)$ admits a basis consisting of trapezohedral paths $\tau_m$ ($m \ge 2$) and their merging images and an explicit construction of such a basis is provided.

Abstract

We study the structure of the space $Ω_3(G)$ of $\partial$-invariant 3-paths in a directed graph $G$. We prove that $Ω_3(G)$ admits a basis consisting of trapezohedral paths $τ_m$ ($m \ge 2$) 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 $Ω_3(G)$ for any finite digraph.

Figures (4)

• Figure 1: Schematic diagram of $T_m$
• Figure 2: Schematic diagram of $f : T_2 \to G$ in Lemma\ref{['lem:image1']}
• Figure 3: Schematic diagram of $f : T_{m+2} \to G$ in Lemma\ref{['lem:image2']}
• Figure 4: Schematic diagram of $f : T_{m+2} \to G$ in Lemma\ref{['lem:image3']}

Theorems & Definitions (63)

• Theorem 1.1
• Definition 2.1: Digraphs
• Definition 2.2: Neighborhood
• Definition 2.3: Induced subgraphs
• Definition 2.4: Induced digraph from $A$ to $B$
• Definition 2.5: Elementary $p$-paths
• Definition 2.6: Boundary operator
• Definition 2.7: Regular $p$-paths
• Definition 2.8: Allowed elementary $p$-paths
• Definition 2.9: $\partial$-invariant $p$-paths