Hypergraphs: connection and separation

TL;DR

This paper explores fundamental connectivity properties of hypergraphs, focusing on cut edges, vertices, and blocks, and establishes relationships between hypergraph decompositions and their incidence graphs.

Contribution

It introduces new results on hypergraph connectivity and relates hypergraph block decompositions to those of their incidence graphs.

Findings

Characterization of cut edges and vertices in hypergraphs

Relationship between hypergraph blocks and incidence graph blocks

New theorems on hypergraph connectivity properties

Abstract

In this paper we study fundamental connectivity properties of hypergraphs from a graph-theoretic perspective, with the emphasis on cut edges, cut vertices, and blocks. To prepare the ground, we define various types of subhypergraphs, as well as various types of walks in a hypergraph. We then prove a number of new results involving cut edges, cut vertices, and blocks. In particular, we describe the exact relationship between the block decomposition of a hypergraph and the block decomposition of its incidence graph.