Linear extension diameter of level induced subposets of the Boolean lattice

The linear extension diameter of a finite poset P is the diameter of the graph on all linear extensions of P as vertices, two of them being adjacent whenever they differ in a single adjacent transposition. We determine the linear extension diameter of the subposet of the Boolean lattice induced by the 1 st and k th levels and describe an explicit construction of all diametral pairs. This partially solves a question of Felsner and Massow. The diametral pairs are obtained from minimal vertex-edge covers of so called dependency graphs, a new concept which may be of independent interest.