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

Hasebe and Tsujie characterized the set of induced N-free and bowtie-free posets as a certain class of recursively defined subposets which they term “ $\mathcal {V}$ -posets”. Here we offer a new characterization of $\mathcal {V}$ -posets by introducing a property we refer to as autonomy. A poset $\mathcal {P}$ is said to be autonomous if there exists a directed acyclic graph D (with adjacency matrix U) whose transitive closure is $\mathcal {P}$ , with the property that any total ordering of the vertices of D so that Gaussian elimination of UTU proceeds without row swaps is a linear extension of $\mathcal {P}$ . Autonomous posets arise from the theory of pressing sequences in graphs, a problem with origins in computational evolutionary biology. 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.