The vertex sets of subtrees of a tree

TL;DR

This paper characterizes when a family of subsets of a set can be realized as subtrees of a tree, establishing necessary and sufficient conditions involving the Helly property and chordality of the intersection graph.

Contribution

It proves that the Helly property and chordality are both necessary and sufficient conditions for such a realization, extending to infinite cases under certain restrictions.

Findings

Helly property and chordality are necessary for subtree realizations.

These conditions are sufficient in finite cases.

Sufficiency extends to some infinite cases with restrictions.

Abstract

Let $\mathcal{F}$ be a set of subsets of a set $W$. When is there a tree $T$ with vertex set $W$ such that each member of $\mathcal{F}$ is the set of vertices of a subtree of $T$? It is necessary that $\mathcal{F}$ has the Helly property and the intersection graph of $\mathcal{F}$ is chordal. We will show that these two necessary conditions are together sufficient in the finite case, and more generally, they are sufficient if no element of $W$ belongs to infinitely many infinite sets in $\mathcal{F}$.