  The holographic entropy cone (HEC) is a polyhedral cone first introduced in
the study of a class of quantum entropy inequalities. It admits a
graph-theoretic description in terms of minimum cuts in weighted graphs, a
characterization which naturally generalizes the cut function for complete
graphs. Unfortunately, no complete facet or extreme-ray representation of the
HEC is known. In this work, starting from a purely graph-theoretic perspective,
we develop a theoretical and computational foundation for the HEC. The paper is
self-contained, giving new proofs of known results and proving several new
results as well. These are also used to develop two systematic approaches for
finding the facets and extreme rays of the HEC, which we illustrate by
recomputing the HEC on $5$ terminals and improving its graph description. We
also report on some partial results for $6$ terminals. Some interesting open
problems are stated throughout.
