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.