The Price of Upwardness

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 commo...

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Published in:Discrete Mathematics and Theoretical Computer Science Vol. 27:3; no. Graph Theory; pp. 1 - 30
Main Authors: Angelini, Patrizio, Biedl, Therese, Chimani, Markus, Cornelsen, Sabine, Da Lozzo, Giordano, Hong, Seok-Hee, Liotta, Giuseppe, Patrignani, Maurizio, Pupyrev, Sergey, Rutter, Ignaz, Wolff, Alexander
Format: Journal Article
Language:English
Published: Nancy DMTCS 01.10.2025
Discrete Mathematics & Theoretical Computer Science
Subjects:
computational geometry
discrete mathematics
Graph representations
Graph theory
Mathematical research
Italy
ISSN:1365-8050, 1462-7264, 1365-8050
Online Access:Get full text
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Summary: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. This is the full version of a paper that appeared in the Proc. 32nd Int. Symp. Graph Drawing & Network Visualization (GD 2024)
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ISSN:1365-8050
1462-7264
1365-8050
DOI:10.46298/dmtcs.15222