N-extendible posets, and how to minimize total weighted completion time

P_4-extendible graphs have been introduced by B. Jamison and S. Olariu [Stud. Appl. Math. 81 (1989), no. 1, 79--87, Discrete Appl. Math. 34 (1991), no. 1-3, 151--164, Discrete Appl. Math. 35 (1992), no. 2, 115--129] as one of several tractable generalizations of P_4-free graphs (cographs). A P_4 has a path of length three. A graph G is P_4-extendible, if every vertex set W of G inducing a P_4 has a proper extension, i.e., there is at most one vertex x not in W which belongs to an induced P_4 that shares vertices with W. In this paper we first examine P_4-extendible comparability graphs and exhibit the class of N-extendible posets, a true superset of N-sparse posets. In particular, every poset of size at most five is N-extendible. Second, we solve the single machine scheduling problem of mini mizing total weighted completion time for N-extendible posets in O(n log n) time, thereby improving an O(n^2) result of Schulz for N-sparse posets