Characterization of unlabeled level planar trees

Abstract

Consider a graph G with vertex set V in which each of the n vertices is assigned a number from the set { 1 , … , k } for some positive integer k. This assignment ϕ is a labeling if all k numbers are used. If ϕ does not assign adjacent vertices the same label, then ϕ forms a leveling that partitions V into k levels. If G has a planar drawing in which the y-coordinate of all vertices match their labels and edges are drawn strictly y-monotone, then G is level planar. In this paper, we consider the class of level trees that are level planar regardless of their labeling. We call such trees unlabeled level planar ( ULP). Our contributions are three-fold. First, we describe which trees are ULP and provide linear-time level planar drawing algorithms for any labeling. Second, we characterize ULP trees in terms of forbidden subtrees so that any other tree must contain a subtree homeomorphic to one of these. Third, we provide a linear-time recognition algorithm for ULP trees.