2026-07-23T01:41:58Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:21297 2024-10-28T09:51:05Z ddc:004 open_access

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 Wolff, Alexander upward drawings beyond planarity upward k-planarity upward outer-1-planarity 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 ≥ 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 upward-k-planarity testing 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Patrizio Angelini and Therese Biedl and Markus Chimani and Sabine Cornelsen and Giordano Da Lozzo and Seok-Hee Hong and Giuseppe Liotta and Maurizio Patrignani and Sergey Pupyrev and Ignaz Rutter and Alexander Wolff 2024 Is Part Of LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.GD.2024.13 urn:nbn:de:0030-drops-212977 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.13 eng https://creativecommons.org/licenses/by/4.0/legalcode