  A jump is a pair of consecutive elements in an extension of a poset which are
incomparable in the original poset. The arboreal jump number is an NP-hard
problem that aims to find an arboreal extension of a given poset with minimum
number of jumps. The contribution of this paper is twofold: (i)~a
characterization that reveals a relation between the number of jumps of an
arboreal order extension and the size of a partition of its elements that
satisfy some structural properties of the covering graph; (ii)~a compact
integer programming model and a heuristic to solve the arboreal jump number
problem along with computational results comparing both strategies. The exact
method provides an optimality certificate for 18 out of 41 instances with
execution time limited to two hours. Furthermore, our heuristic was able to
find good feasible solutions for all instances in less than three minutes.
