, Volume 30, Issue 1, pp 339–350 | Cite as

The Arboreal Jump Number of an Order

  • Adriana P. Figueiredo
  • Michel Habib
  • Sulamita Klein
  • Jayme Luiz Szwarcfiter


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.


Arboreal jump number Order extensions Greedy chain decompositions Jump number Partially ordered sets 


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Adriana P. Figueiredo
    • 1
  • Michel Habib
    • 2
    • 3
  • Sulamita Klein
    • 4
    • 5
  • Jayme Luiz Szwarcfiter
    • 4
    • 5
    • 6
  1. 1.DMEUniversidade Federal do Estado do Rio de JaneiroRio de JaneiroBrazil
  2. 2.LIAFAUniversité Paris Diderot - Paris 7ParisFrance
  3. 3.CNRSUniversité Paris Diderot - Paris 7ParisFrance
  4. 4.IMUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  5. 5.COPPEUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  6. 6.NCEUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

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