It has recently been shown that the edges of the characteristic imset polytope $\operatorname{CIM}_p$ have strong links to causality discovery, as many algorithms can interpret them as greedily constrained edge walks. was shown. is known. To better understand the general edge structure of polytopes, we describe the facet edge structure with a clear combinatorial interpretation. For any undirected graph $G$ there is a face $\operatorname{CIM}_G$ that is the convex hull of the property imset. of a DAG with a skeleton $G$. If $G$ is a tree, we do a complete edge description of $\operatorname{CIM}_G$, leading to interesting connections to other polytopes. In particular, if $G$ is a tree, well-studied stable set polytopes can be recovered as faces of $\operatorname{CIM}_G$. Based on this connection, if $G$ is a cycle, we can also describe all edges of $\operatorname{CIM}_G$, suggesting a potential generalization. Next, we present an algorithm that uses newly discovered edges to learn directed trees from data. This outperforms traditional methods on simulated Gaussian data.
