Level planar embedding in linear time

Abstract

In a level directed acyclic graph G = (V;E) the vertex set V is partitioned into k ≤ |V| levels V1; V2;:: Vksuch that for each edge (u; v) Є E with u Є Viand v Є Vjwe have i < j. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level Vi, all v Є Viare drawn on the line li= {(x; k − i) | x Є ℝ}, the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to draw a level planar graph without edge crossings, a level planar embedding of the level graph has to be computed. Level planar embeddings are characterized by linear orderings of the vertices in each Vi(1 ≤ i ≤ k). We present an O(|V|) time algorithm for embedding level planar graphs. This approach is based on a level planarity test by Jünger, Leipert, and Mutzel [6].