Minimizing the jump number for partially-ordered sets: a graph-theoretic approach, II

Abstract

This paper is a continuation of another author's work (Order 1 (1985) 7–19), where arc diagrams of posets have been successfully applied to solve the jump number problem for N-free posets. Here, we consider arbitrary posets and, again making use of arc diagrams of posets, we define two special types of greedy chains: strongly and semi-strongly greedy. Every strongly greedy chain may begin an optimal linear extension (Theorem 1 and Corollary 1). If a poset has no strongly greedy chains, then it has an optimal linear extension which starts with a semi-strongly greedy chain (Theorem 2). Therefore, every poset has an optimal linear extension which consists entirely of strongly and semi-strongly greedy chains. This fact leads to a polynomial-time algorithm for the jump number problem in the class of posets whose arc diagrams contain a bounded number of dummy arcs.