Characterization of Unlabeled Level Planar Trees

Abstract

Consider a graph G drawn in the plane so that each vertex lies on a distinct horizontal line lj = {(x, j) | x ∈ R}. The bijection φ that maps the set of n vertices V to a set of distinct horizontal lines lj forms a labeling of the vertices. Such a graph G with the labeling φ is called an n-level graph and is said to be n-level planar if it can be drawn with straight-line edges and no crossings while keeping each vertex on its own level. In this paper, we consider the class of trees that are n-level planar regardless of their labeling. We call such trees unlabeled level planar (ULP). Our contributions are three-fold. First, we provide a complete characterization of ULP trees in terms of a pair of forbidden subtrees. Second, we show how to draw ULP trees in linear time. Third, we provide a linear time recognition algorithm for ULP trees.