K-Level Crossing Minimization Is Np-Hard For Trees

The k-level crossing minimization problem for graphs has received much interest in the graph drawing literature. In this paper we focus on the special case of trees. We show that the 2-level crossing minimization problem for trees where the order of the vertices on one level is fixed is solvable in quadratic time. We also show that the k-level crossing minimization problem for trees for an arbitrary number of levels is NP-Hard. This result exposes a source of difficulty for algorithm designers that compounds earlier results relating to the 2-level crossing minimization problem for graphs.