A New Characterization of $\mathcal{V}$-Posets

TL;DR

This paper introduces a new way to characterize $alV$-posets using the concept of autonomy, linking them to directed acyclic graphs and pressing sequences in graphs, with implications for enumeration and recognition algorithms.

Contribution

It presents a novel characterization of $alV$-posets through the property of autonomy, connecting poset theory with graph pressing sequences and providing recognition algorithms.

Findings

Autonomous posets are characterized by the existence of a DAG with specific properties.

Pressing sequences in graphs can be partitioned into families corresponding to autonomous posets.

An efficient algorithm for recognizing autonomous posets is developed.

Abstract

In 2016, Hasebe and Tsujie gave a recursive characterization of the set of induced $N$-free and bowtie-free posets; Misanantenaina and Wagner studied these orders further, naming them "$\mathcal{V}$-posets". Here we offer a new characterization of $\mathcal{V}$-posets by introducing a property we refer to as autonomy. A poset $\cP$ is said to be autonomous if there exists a directed acyclic graph $D$ (with adjacency matrix $U$) whose transitive closure is $\cP$, with the property that any total ordering of the vertices of $D$ so that Gaussian elimination of $U^TU$ proceeds without row swaps is a linear extension of $\cP$. Autonomous posets arise from the theory of pressing sequences in graphs, a problem with origins in phylogenetics. The pressing sequences of a graph can be partitioned into families corresponding to posets; because of the interest in enumerating pressing sequences, we investigate when this partition has only one block, that is, when the pressing sequences are all linear extensions of a single autonomous poset. We also provide an efficient algorithm for recognition of autonomy using structural information and the forbidden subposet characterization, and we discuss a few open questions that arise in connection with these posets.