The Arboreal Jump Number of an Order

Abstract

Let \({\mathcal P}\) be a partial order and \({\mathcal A}\) an arboreal extension of it (i.e. the Hasse diagram of \({\mathcal A}\) is a rooted tree with a unique minimal element). A jump of \({\mathcal A}\) is a relation contained in the Hasse diagram of \({\mathcal A}\), but not in the order \({\mathcal P}\). The arboreal jump number of \({\mathcal A}\) is the number of jumps contained in it. We study the problem of finding the arboreal extension of \({\mathcal P}\) having minimum arboreal jump number—a problem related to the well-known (linear) jump number problem. We describe several results for this problem, including NP-completeness, polynomial time solvable cases and bounds. We also discuss the concept of a minimal arboreal extension, namely an arboreal extension whose removal of one jump makes it no longer arboreal.