  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.
