Upward Planar Drawing of Single-Source Acyclic Digraphs

An upward plane drawing of a directed acyclic graph is a plane drawing of the digraph in which each directed edge is represented as a curve monotone increasing in the vertical direction. Thomassen has given a nonalgorithmic, graph-theoretic characterization of those directed graphs with a single source that admit an upward plane drawing. This paper presents an efficient algorithm to test whether a given single-source acyclic digraph has an upward plane drawing and, if so, to find a representation of one such drawing. This result is made more significant in light of the recent proof by Garg and Tamassia that the problem is NP-complete for general digraphs. The algorithm decomposes the digraph into biconnected and triconnected components and defines conditions for merging the components into an upward plane drawing of the original digraph. To handle the triconnected components, we provide a linear algorithm to test whether a given plane drawing of a single-source digraph admits an upward plane drawing with the same faces and outer face, which also gives a simpler, algorithmic proof of Thomassen's result. The entire testing algorithm (for general single-source directed acyclic graphs) operates in $O(n^2)$ time and $O(n)$ space ($n$ being the number of vertices in the input digraph) and represents the first polynomial-time solution to the problem.