Generating Linear Extensions by Adjacent Transpositions

If P is a rooted forest in which no element has exactly one child, then the linear extensions of P can be successively generated by transpositions of consecutive elements; i.e., P is adjacent-traversable. More generally, if P is adjacent-traversable and P′ is obtained from P by adding an antichain of size at least two covering (or covered by) a single element of P, then P′ is also adjacent-traversable. The proof is constructive.