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.
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S. Klein and J. L. Szwarcfiter were partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro, FAPERJ, Brazil.
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Figueiredo, A.P., Habib, M., Klein, S. et al. The Arboreal Jump Number of an Order. Order 30, 339–350 (2013). https://doi.org/10.1007/s11083-012-9246-4
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DOI: https://doi.org/10.1007/s11083-012-9246-4