Oriented Hypergraphs I: Introduction and Balance

TL;DR

This paper introduces oriented hypergraphs, extending signed graphs, and classifies their balanced circuits, providing foundational structures for further combinatorial and algebraic studies.

Contribution

It extends the concept of signed graphs to oriented hypergraphs and classifies their balanced circuits comprehensively.

Findings

Complete classification of balanced circuits in oriented hypergraphs

Decomposition of oriented hypergraphs into balanced, balanceable, and unbalanceable families

Extension of circuit classification from signed graphs to hypergraphs

Abstract

An oriented hypergraph is an oriented incidence structure that extends the concept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented hypergraphs. Oriented hypergraphs are further decomposed into three families -- balanced, balanceable, and unbalanceable -- and we obtain a complete classification of the balanced circuits of oriented hypergraphs.