We define treetopes, a generalization of the three-dimensional roofless polyhedra (Halin graphs) to arbitrary dimensions. Like roofless polyhedra, treetopes have a designated base facet which intersects every face of dimension greater than one in more than one point. We prove an equivalent characterization of the 4-treetopes using the concept of clustered planarity from graph drawing, and we use this characterization to recognize the graphs of 4-treetopes in polynomial time. This result provides one of the first classes of 4-polytopes, other than pyramids and stacked polytopes, that can be recognized efficiently from their graphs. Additionally we show that every d -dimensional treetope (with $$d\ge 3$$ d ≥ 3 ) has at most $$d+1$$ d + 1 base facets, and that despite not having any forbidden minors the 4-treetopes obey a separator theorem like the one for planar graphs.

中文翻译：

---

树梢及其图形

我们定义树梢，将三维无屋顶多面体（Halin 图）推广到任意维度。像无屋顶的多面体一样，树梢有一个指定的基面，它与尺寸大于一个点的每个面相交。我们使用图形绘制中的聚类平面性概念证明了 4-treetopes 的等效表征，并且我们使用这种表征在多项式时间内识别 4-treetopes 的图形。该结果提供了第一类 4-polytopes 之一，除了金字塔和堆叠多面体，可以从它们的图中有效地识别。此外，我们表明每个 d 维树梢（具有 $$d\ge 3$$ d ≥ 3 ）至多具有 $$d+1$$ d + 1 个基本面，