Minimally Connected Hypergraphs

TL;DR

This paper introduces a construction process and a numerical invariant called tightness for minimally connected hypergraphs, establishing their fundamental properties, bounds on chromatic numbers, and algorithms for identifying spanning subhypergraphs.

Contribution

It defines a new construction process and invariant for minimally connected hypergraphs, proving key properties and providing algorithms for spanning subhypergraph identification.

Findings

Established properties of minimally connected hypergraphs.

Provided bounds on chromatic numbers.

Developed a polynomial-time algorithm for spanning subhypergraph identification.

Abstract

Graphs and hypergraphs are foundational structures in discrete mathematics. They have many practical applications, including the rapidly developing field of bioinformatics, and more generally, biomathematics. They are also a source of interesting algorithmic problems. In this paper, we define a \textit{construction process} for minimally connected $r$-uniform hypergraphs, which captures the intuitive notion of building a hypergraph piece-by-piece, and a numerical invariant called the \textit{tightness}, which is independent of the construction process used. Using these tools, we prove some fundamental properties of minimally connected hypergraphs. We also give bounds on their chromatic numbers and provide some results involving edge colorings. We show that every connected $r$-uniform hypergraph contains a minimally connected spanning subhypergraph and provide a polynomial-time algorithm for identifying such a subhypergraph.