Hypertrees and their host trees: a survey

Abstract

A hypergraph $\mathcal{H}=(V,\mathcal{E})$ is a hypertree if it admits a tree $T$ with vertex set $V$ such that every edge of $\mathcal{H}$ induces a subtree of $T$. A tree like that is called a host tree. Several characterizations and properties of hypertrees have been discovered over the years. However, the interest in the structure of their host trees was weaker and restricted to particular scenarios where they arise, like the clique tree of chordal graphs. In that special case, the proofs of most characteristics of clique trees that exist in the literature rely significantly on the structural properties of chordal graphs. The purpose of this work is the study of the properties of the host trees of hypertrees in a more general context and have them described in a single place, giving simpler proofs for known facts, generalizing others and introducing some new concepts that the author considers that are relevant for the study of the topic. Particularly, we will determine what edges can be found in some host tree of a hypertree, and how these edges must be combined to form a host tree, with an emphasis in tools like the basis and the completion of a hypergraph, and the concept of equivalent hypergraphs.