Characterizing and transforming DAGs within the ℐ-lca framework

We explore the connections between clusters and least common ancestors (LCAs) in directed acyclic graphs (DAGs), focusing on the interplay between so-called $ℐ$-lca-relevant DAGs and DAGs with the $ℐ$-lca-property. Here, $ℐ$ denotes a set of integers. In $ℐ$-lca-relevant DAGs, each vertex is the unique LCA for some subset $A$ of leaves of size $\left|A\right|\in ℐ$, whereas in a DAG with the $ℐ$-lca-property there exists a unique LCA for every subset $A$ of leaves satisfying $\left|A\right|\in ℐ$. We elaborate on the difference between these two properties and establish their close relationship to pre-$ℐ$-ary and $ℐ$-ary set systems. This, in turn, generalizes results established for (pre-) binary and $k$-ary set systems. Moreover, we build upon recently established results that use a simple operator $\ominus$, enabling the transformation of arbitrary DAGs into $ℐ$-lca-relevant DAGs. This process reduces unnecessary complexity while preserving key structural properties of the original DAG. The set ${ℭ}_G$ consists of all clusters in a DAG $G$, where clusters correspond to the descendant leaves of vertices. While in some cases ${ℭ}_H={ℭ}_G$ when transforming $G$ into an $ℐ$-lca-relevant DAG $H$, it often happens that certain clusters in ${ℭ}_G$ do not appear as clusters in $H$. To understand this phenomenon in detail, we characterize the subset of clusters in ${ℭ}_G$ that remain in $H$ for DAGs $G$ with the $ℐ$-lca-property. Furthermore, we show that the set $W$ of vertices required to transform $G$ into $H=G\ominus W$ is uniquely determined for such DAGs. This, in turn, allows us to show that the “shortcut-free” version of the transformed DAG $H$ is always a tree or a galled-tree whenever ${ℭ}_G$ represents the clustering system of a tree or galled-tree and $G$ has the $ℐ$-lca-property. In the latter case ${ℭ}_H={ℭ}_G$ always holds.