A characterization of positive spanning sets with ties to strongly connected digraphs

TL;DR

This paper explores the relationship between positive spanning sets and strongly connected digraphs, introducing a decomposition structure inspired by digraph ear decomposition to better understand PSSs.

Contribution

It establishes a novel connection between positive spanning sets and strongly connected digraphs, providing a new decomposition framework for PSSs.

Findings

Positive spanning sets can be viewed as a generalization of strongly connected digraphs.

A new decomposition structure for PSSs is proposed based on digraph ear decomposition.

The work offers insights into the structure of PSSs, advancing their theoretical understanding.

Abstract

Positive spanning sets (PSSs) are families of vectors that span a given linear space through non-negative linear combinations. Despite certain classes of PSSs being well understood, a complete characterization of PSSs remains elusive. In this paper, we explore a relatively understudied relationship between positive spanning sets and strongly edge-connected digraphs, in that the former can be viewed as a generalization of the latter. We leverage this connection to define a decomposition structure for positive spanning sets inspired by the ear decomposition from digraph theory.