Distance ideals of digraphs

TL;DR

This paper studies the algebraic properties of distance matrices in strongly connected digraphs, characterizes a specific family with trivial distance ideals, and explores the complexity of their structural classification.

Contribution

It introduces the concept of patterns to characterize digraphs with a single trivial distance ideal and generalizes known results from undirected graphs to directed graphs.

Findings

Circuits with 3 vertices are in the family .

Complete graphs and bipartite graphs are included in the family .

Circuits are minimal forbidden induced subdigraphs for the family .

Abstract

We focus on strongly connected, strong for short, digraphs since in this setting distance is defined for every pair of vertices. Distance ideals generalize the spectrum and Smith normal form of several distance matrices associated with strong digraphs. We introduce the concept of pattern which allow us to characterize the family $\Gamma_1$ of digraphs with only one trivial distance ideal over ${\mathbb Z}$. This result generalizes an analogous result for undirected graphs that states that connected graphs with one trivial ideal over $\mathbb{Z}$ consists of either complete graphs or complete bipartite graphs. It turns out that the strong digraphs in $\Gamma_1$ consists in the circuit with 3 vertices and a family $\Lambda$ of strong digraphs that contains complete graphs and complete bipartite graphs, regarded as digraphs. We also compute all distance ideals of some strong digraphs in the family $\Lambda$. Then, we explore circuits, which turn out to be an infinite family of minimal forbidden digraphs, as induced subdigraphs, for the strong digraphs in $\Gamma_1$. This suggests that a characterization of $\Gamma_1$ in terms of forbidden induced subdigraphs is harder than using patterns.