{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T11:33:21Z","timestamp":1760441601004,"version":"3.41.0"},"reference-count":23,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2019,10,4]],"date-time":"2019-10-04T00:00:00Z","timestamp":1570147200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Algorithms"],"published-print":{"date-parts":[[2019,10,31]]},"abstract":"<jats:p>\n            We introduce and study the problem Ordered Level Planarity, which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a\n            <jats:italic>y<\/jats:italic>\n            -monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems.\n          <\/jats:p>\n          <jats:p>\n            Geodesic Planarity asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge\n            <jats:italic>e<\/jats:italic>\n            is realized as a polygonal path\n            <jats:italic>p<\/jats:italic>\n            composed of line segments with two adjacent directions from a given set\n            <jats:italic>S<\/jats:italic>\n            of directions that is symmetric with respect to the origin. Our results on Ordered Level Planarity imply\n            <jats:italic>NP<\/jats:italic>\n            -hardness for any\n            <jats:italic>S<\/jats:italic>\n            with \u2223S\u2223 \u2265 4, even if the given graph is a matching. Manhattan Geodesic Planarity is the special case where\n            <jats:italic>S<\/jats:italic>\n            contains precisely the horizontal and vertical directions. Katz, Krug, Rutter, and Wolff claimed that Manhattan Geodesic Planarity can be solved in polynomial time for the special case of matchings [GD\u201909]. Our results imply that this is incorrect unless\n            <jats:italic>P<\/jats:italic>\n            =\n            <jats:italic>NP<\/jats:italic>\n            . Our reduction extends to settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer, and \u0160tefankovi\u010d.\n          <\/jats:p>\n          <jats:p>Ordered Level Planarity turns out to be a special case of T-Level Planarity, Clustered Level Planarity, and Constrained Level Planarity. Thus, our results strengthen previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two non-trivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati, and Roselli.<\/jats:p>","DOI":"10.1145\/3359587","type":"journal-article","created":{"date-parts":[[2019,10,7]],"date-time":"2019-10-07T12:20:59Z","timestamp":1570450859000},"page":"1-25","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Ordered Level Planarity and Its Relationship to Geodesic Planarity, Bi-Monotonicity, and Variations of Level Planarity"],"prefix":"10.1145","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4532-3765","authenticated-orcid":false,"given":"Boris","family":"Klemz","sequence":"first","affiliation":[{"name":"Freie Universit\u00e4t Berlin, Takustra\u00dfe, Berlin, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0351-5945","authenticated-orcid":false,"given":"G\u00fcnter","family":"Rote","sequence":"additional","affiliation":[{"name":"Freie Universit\u00e4t Berlin, Takustra\u00dfe, Berlin, Germany"}]}],"member":"320","published-online":{"date-parts":[[2019,10,4]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-50106-2_37"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2014.12.019"},{"key":"e_1_2_1_3_1","first-page":"185","article-title":"Quasi-parallel segments and characterization of unique bichromatic matchings","volume":"6","author":"Asinowski Andrei","year":"2015","journal-title":"J. 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