Diametral pairs of linear extensions

Given a finite poset $\mathcal{P}$, we consider pairs of linear extensions of $\mathcal{P}$ with maximal distance, where the distance between two linear extensions $L_1, L_2$ is the number of pairs of elements of $\mathcal{P}$ appearing in different orders in $L_1$ and $L_2$. A diametral pair maximizes the distance among all pairs of linear extensions of $\mathcal{P}$. Felsner and Reuter defined the linear extension diameter of $\mathcal{P}$ as the distance between a diametral pair of linear extensions. We show that computing the linear extension diameter is NP-complete in general but can be solved in polynomial time for posets of width 3. Felsner and Reuter conjectured that, in every diametral pair, at least one of the linear extensions reverses a critical pair. We construct a counterexample to this conjecture. On the other hand, we show that a slightly stronger property holds for many classes of posets: we call a poset diametrally reversing if, in every diametral pair, both linear extensions reverse a critical pair. Among other results we show that interval orders and 3-layer posets are diametrally reversing. From the latter it follows that almost all posets are diametrally reversing.