Interval orders without odd crowns are defect optimal

Abstract

The defect of a (partial) order relationP is defined to be the rank of the kernel of the associated incidence matrix. Gierz and Poguntke [7] have shown that the defect provides a lower bound for the number of incomparable adjacent pairs in an arbitrary topological sorting ofP. We show that this bound is sharp for interval orders without odd crowns. Furthermore, an efficient algorithm for topological sortings of such orders is presented which achieves the bound. We finally exhibit a natural matroid structure associated with the optimal topological sortings under consideration, which permits to solve the weighted case.