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The Price of Upwardness
Angelini, Patrizio
Biedl, Therese Chimani, Markus Cornelsen, Sabine Da Lozzo, Giordano Hong, Seok-Hee Liotta, Giuseppe Patrignani, Maurizio Pupyrev, Sergey Rutter, Ignaz
Biedl, Therese
Chimani, Markus
Cornelsen, Sabine
Da Lozzo, Giordano
Hong, Seok-Hee
Liotta, Giuseppe
Patrignani, Maurizio
Pupyrev, Sergey
Rutter, Ignaz
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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. 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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Date
2025
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Keywords
Upward drawings, Beyond planarity, Upward k-planarity, Upward outer-1-planarity
Citation
Angelini, Patrizio, Therese Biedl, Markus Chimani, et al. “The Price of Upwardness.” Discrete Mathematics and Theoretical Computer Science 27 (3): 1–30. 2025.
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Attribution 4.0 International