The Price of Upwardness

TL;DR

This paper explores upward k-planar drawings of DAGs, analyzing their properties, computational complexity, and providing algorithms for specific cases, extending the understanding of upward planarity beyond traditional planarity constraints.

Contribution

It introduces the concept of upward k-planar drawings, analyzes their properties for various DAG classes, and establishes the NP-completeness of testing upward-k-planarity even for k=1.

Findings

Crossings per edge are unbounded for some DAG classes.

Testing upward-1-planarity is NP-complete.

Linear-time algorithm for outerpath DAGs with all vertices on the outer face.

Abstract

Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward $k$-planar drawings of DAGs in which the edges are monotonically increasing in a common direction and every edge is crossed at most $k$ times for some integer $k \ge 1$. We show that the number of crossings per edge in a monotone drawing is in general unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth DAGs. However, it is at most two for outerpaths and it is at most quadratic in the bandwidth in general. From the computational point of view, we prove that testing upward-$k$-planarity is NP-complete already for $k=1$ and even for restricted instances for which upward planarity testing is polynomial. On the positive side, we can decide in linear time whether a single-source DAG admits an upward 1-planar drawing in which all vertices are incident to the outer face.