Containment Graphs and Posets of Paths in a Tree: Wheels and Partial Wheels

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_2k(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.