  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.
