A Characterization of Partially Ordered Sets with Linear Discrepancy Equal to 2

Abstract

The linear discrepancy of a poset P is the least k such that there is a linear extension L of P such that if x and y are incomparable in P, then |hL(x)–hL(y)|≤k, where hL(x) is the height of x in L. Tanenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem of characterizing the posets of linear discrepancy 2. We show that this problem is equivalent to finding the posets with linear discrepancy equal to 3 having the property that the deletion of any point results in a reduction in the linear discrepancy. Howard determined that there are infinitely many such posets of width 2. We complete the forbidden subposet characterization of posets with linear discrepancy equal to 2 by finding the minimal posets of width 3 with linear discrepancy equal to 3. We do so by showing that, with a small number of exceptions, they can all be derived from the list for width 2 by the removal of specific comparisons.