Embracing exchange sequences and oriented matroid polyhedron diameter

Abstract

We reduce the embracing exchange distance of bases of oriented matroids to the metric of oriented matroid polyhedra. This allows us to disprove recent conjectures of Caoduro, Khodamoradi, Paat, and Shepherd and of Bérczi and Nádor. On the other hand, we show that any two embracing bases of an oriented matroid of rank $r$ can be transformed into each other in at most $2r^{\log_2(r)+3}$ steps and in at most $r$ steps in a graphic oriented matroid or a Lawrence oriented matroid, thus confirming the conjecture in these cases.