标题： 树的子树的顶点集 显示英文标题

标题： The vertex sets of subtrees of a tree

摘要： 设 $\mathcal{F}$ 是集合 $W$ 的子集组成的集合。何时存在一棵顶点集为 $W$ 的树 $T$，使得 $\mathcal{F}$ 的每个成员都是 $T$ 的某个子树的顶点集？ 有必要让 $\mathcal{F}$ 具有 Helly 性质，并且 $\mathcal{F}$ 的交图是弦图。 我们将证明这两个必要条件在有限情况下是充分的，并且更一般地，如果 $W$ 的每个元素不属于 $\mathcal{F}$ 中的无穷多个无限集，则它们是充分的。 显示英文摘要

摘要： 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 $\mathcal{F}$.