Oriented Hypergraphs: Balanceability

TL;DR

This paper introduces the concept of oriented hypergraphs and generalizes circuit classification from signed graphs to these structures, providing new tools for analyzing balanceability and matroid representations.

Contribution

It extends the theory of signed graphs to oriented hypergraphs, generalizes Camion's algorithm, and characterizes unbalanceable circuits in these structures.

Findings

Generalization of circuit classification to oriented hypergraphs

Application of Camion's algorithm to hypergraphs

Characterization of unbalanceable circuits in matroids

Abstract

An oriented hypergraph is an oriented incidence structure that extends the concepts of signed graphs, balanced hypergraphs, and balanced matrices. We introduce hypergraphic structures and techniques that generalize the circuit classification of the signed graphic frame matroid to any oriented hypergraphic incidence matrix via its locally-signed-graphic substructure. To achieve this, Camion's algorithm is applied to oriented hypergraphs to provide a generalization of reorientation sets and frustration that is only well-defined on balanceable oriented hypergraphs. A simple partial characterization of unbalanceable circuits extends the applications to representable matroids demonstrating that the difference between the Fano and non-Fano matroids is one of balance.