Characterizing and Transforming DAGs within the I-LCA Framework

TL;DR

This paper investigates the properties of DAGs related to least common ancestors and clusters, introduces transformations to simplify DAGs while preserving key structures, and characterizes when clusters are retained or lost during these transformations.

Contribution

It generalizes existing results for binary and k-ary set systems to broader DAG classes and introduces a transformation method that preserves essential clustering properties.

Findings

Transformation reduces DAG complexity while maintaining key structures.

Characterization of clusters retained after transformation.

Transformed DAGs are trees or galled-trees under certain conditions.

Abstract

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 $|A|\in I$, whereas in a DAG with the $I$-lca-property there exists a unique LCA for every subset $A$ of leaves satisfying $|A|\in I$. 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.