Abstract

A level graph G = (V; E; ) is a directed acyclic graph with a mapping : V ! f1; 2; : : : ; kg, k 1, that partitions the vertex set V as V = V 1 [V 2 [: : :[V k , V j = 1 (j), V i \V j = ; for i 6= j, such that (v) = (u)+1 for each edge (u; v) 2 E. The graph G is level planar if it can be drawn in the plane such that for each level V i , all v 2 V i are drawn on the line l i = f(x; k i) j x 2 Rg, the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In this paper we give a characterization of level planar graphs in terms of minimal forbidden subgraphs called minimal level non-planar subgraph patterns (MLNP). We show that a MLNP is completely characterized by either a tree, a level non-planar cycle or a level planar cycle with certain path augmentations. These characterizations are an important first step towards attacking the NP-hard level planarization problem.

Citations