A Graph-Theoretic Approach to the Jump-Number Problem

Abstract

The purpose of this paper is to discuss the jump number problem for partially ordered sets (posets) using their arc representations, in which poset elements are assigned to arcs. We present the solution of the jump number problem restricted toN-free posets. The Hasse diagram of anN-free poset is a line digraph, so it has a unique root digraph which is its arc representation. It is shown that the jump number of an i-free poset is equal to the cyclomatic number of its root digraph. We illustrate also that several other conclusions forN-free posets can be easily proved in terms of root digraphs. In particular, we discuss the scope of the greedy algorithm and jump-critical posets. Moreover, we investigate the problem for arbitrary posets by showing that, in the general case, the iump number of a poset is eaual to the cyclomatic number of a certain digraph which can be derived from a poset arc representation. Then, we strengthen the greedy algorithm and exhibit a class of posets for which it generates optimal linear extensions. Finally, we give a short informal survey of construction methods for arc representations of posets and a list of the most important contributions to the jump number problem and construction methods for arc representations.