Hypertrees and their host trees: a survey
Pablo De Caria Di Fonzo
TL;DR
This survey addresses the problem of understanding host trees for hypertrees in a general hypergraph framework, not restricted to chordal or dually chordal graphs. The authors introduce equivalence among hypertrees, along with completion $Comp(\mathcal{H})$ and basis $\mathcal{B}(\mathcal{H})$, and develop a cohesive theory connecting host trees to these constructions with linear-time implications. They characterize which edges can appear in host trees via the intersection function $I_{\mathcal{H}}(uv)$ and the 2-section of $\overline{\mathcal{H}_{uv}}$, and show that the basic sets fully govern host-tree structure. A central result is that host trees are precisely the maximum weight spanning trees of a complete graph on $V(\mathcal{H})$ with weights $w(uv)=|\{F\in E(\mathcal{H}): \{u,v\}\subseteq F\}|$, with equality conditions tying to the hypertree condition; special cases recover clique trees for chordal graphs and compatible trees for dually chordal graphs. The work provides a unified, toolkit-based view with simpler proofs and introduces new concepts like basic hypertrees, offering a foundation for further hypergraph-based representations of graph structures.
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.