We consider questions regarding the containment graphs of paths in a tree (CPT graphs), a subclass of comparability graphs, and the containment posets of paths in a tree (CPT orders). In 1984, Corneil and Golumbic observed that a graph G may be CPT, yet not every transitive orientation of G necessarily has a CPT representation, illustrating this on the even wheels W 2 k ( k ≥ 3). Motivated by this example, we characterize the partial wheels that are containment graphs of paths in a tree, and give a number of examples and obstructions for this class. Our characterization gives the surprising result that all partial wheels that admit a transitive orientation are CPT graphs. We then characterize the CPT orders whose comparability graph is a partial wheel.

中文翻译：

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树中路径的包含图和位姿：轮子和部分轮子

我们考虑有关树中路径的包含图（CPT 图）、可比性图的子类和树中路径的包含组（CPT 顺序）的问题。1984 年，Corneil 和 Golumbic 观察到图 G 可能是 CPT，但并非 G 的每个传递方向都必须具有 CPT 表示，在偶数轮 W 2 k ( k ≥ 3) 上说明了这一点。受这个例子的启发，我们描述了作为树中路径包含图的部分轮子，并给出了这个类的一些例子和障碍物。我们的表征给出了令人惊讶的结果，即所有允许传递方向的部分轮都是 CPT 图。然后，我们描述了可比性图是部分轮的 CPT 订单。