Proper and planar drawings of graphs on three layers

Abstract

A graph is proper k-layer planar, for an integer k ≥ 0, if it admits a planar drawing in which the vertices are drawn on k horizontal lines called layers and each edge is drawn as a straight-line segment between end-vertices on adjacent layers. In this paper, we point out errors in an algorithm of Fößmeier and Kaufmann (CIAC, 1997) for recognizing proper 3-layer planar graphs, and then present a new characterization of this set of graphs that is partially based on their algorithm. Using the characterization, we then derive corresponding linear-time algorithms for recognizing and drawing proper 3-layer planar graphs. On the basis of our results, we predict that the approach of Fößmeier and Kaufmann will not easily generalize for drawings on four or more layers and suggest another possible approach along with some of the reasons why it may be more successful. © Springer-Verlag Berlin Heidelberg 2005.