{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:41:57Z","timestamp":1740109317197,"version":"3.37.3"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2023,3,11]],"date-time":"2023-03-11T00:00:00Z","timestamp":1678492800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,3,11]],"date-time":"2023-03-11T00:00:00Z","timestamp":1678492800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002345","name":"Eberhard Karls Universit\u00e4t T\u00fcbingen","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100002345","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Algorithmica"],"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Bitonic <jats:italic>st<\/jats:italic>-orderings for <jats:italic>st<\/jats:italic>-planar graphs were introduced as a method to cope with several graph drawing problems. Notably, they have been used to obtain the best-known upper bound on the number of bends for upward planar polyline drawings with at most one bend per edge in polynomial area. For an <jats:italic>st<\/jats:italic>-planar graph that does not admit a bitonic <jats:italic>st<\/jats:italic>-ordering, one may split certain edges such that for the resulting graph such an ordering exists. Since each split is interpreted as a bend, one is usually interested in splitting as few edges as possible. While this optimization problem admits a linear-time algorithm in the fixed embedding setting, it remains open in the variable embedding setting. We close this gap in the literature by providing a linear-time algorithm that optimizes over all embeddings of the input <jats:italic>st<\/jats:italic>-planar graph. The best-known lower bound on the number of required splits of an <jats:italic>st<\/jats:italic>-planar graph with <jats:italic>n<\/jats:italic> vertices is <jats:inline-formula><jats:alternatives><jats:tex-math>$$n-3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. However, it is possible to compute a bitonic <jats:italic>st<\/jats:italic>-ordering without any split for the <jats:italic>st<\/jats:italic>-planar graph obtained by reversing the orientation of all edges. In terms of upward planar polyline drawings in polynomial area, the former translates into <jats:inline-formula><jats:alternatives><jats:tex-math>$$n-3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> bends, while the latter into no bends. We show that this idea cannot always be exploited by describing an <jats:italic>st<\/jats:italic>-planar graph that needs at least <jats:inline-formula><jats:alternatives><jats:tex-math>$$n-5$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mn>5<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> splits in both orientations. We provide analogous bounds for graphs with small degree. Finally, we further investigate the relationship between splits in bitonic st-orderings and bends in upward planar polyline drawings with polynomial area, by providing bounds on the number of bends in such drawings.<\/jats:p>","DOI":"10.1007\/s00453-023-01111-5","type":"journal-article","created":{"date-parts":[[2023,3,11]],"date-time":"2023-03-11T13:02:55Z","timestamp":1678539775000},"page":"2667-2692","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bitonic st-Orderings for Upward Planar Graphs: Splits and Bends in the Variable Embedding Scenario"],"prefix":"10.1007","volume":"85","author":[{"given":"Patrizio","family":"Angelini","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael A.","family":"Bekos","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1441-4189","authenticated-orcid":false,"given":"Henry","family":"F\u00f6rster","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Gronemann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,3,11]]},"reference":[{"issue":"1","key":"1111_CR1","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/BF02122694","volume":"10","author":"H de Fraysseix","year":"1990","unstructured":"de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41\u201351 (1990). https:\/\/doi.org\/10.1007\/BF02122694","journal-title":"Combinatorica"},{"issue":"1","key":"1111_CR2","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1007\/BF02086606","volume":"16","author":"G Kant","year":"1996","unstructured":"Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4\u201332 (1996). https:\/\/doi.org\/10.1007\/BF02086606","journal-title":"Algorithmica"},{"issue":"2","key":"1111_CR3","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1007\/PL00009189","volume":"20","author":"D Harel","year":"1998","unstructured":"Harel, D., Sardas, M.: An algorithm for straight-line drawing of planar graphs. Algorithmica 20(2), 119\u2013135 (1998). https:\/\/doi.org\/10.1007\/PL00009189","journal-title":"Algorithmica"},{"issue":"3","key":"1111_CR4","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/S0925-7721(97)00026-6","volume":"9","author":"TC Biedl","year":"1998","unstructured":"Biedl, T.C., Kant, G.: A better heuristic for orthogonal graph drawings. Comput Geom. 9(3), 159\u2013180 (1998). https:\/\/doi.org\/10.1016\/S0925-7721(97)00026-6","journal-title":"Comput Geom."},{"key":"1111_CR5","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/BF02187706","volume":"1","author":"P Rosenstiehl","year":"1986","unstructured":"Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput. Geom. 1, 343\u2013353 (1986). https:\/\/doi.org\/10.1007\/BF02187706","journal-title":"Discrete Comput. Geom."},{"issue":"3","key":"1111_CR6","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1016\/0304-3975(76)90086-4","volume":"2","author":"S Even","year":"1976","unstructured":"Even, S., Tarjan, R.E.: Computing an st-Numbering. Theor. Comput. Sci. 2(3), 339\u2013344 (1976). https:\/\/doi.org\/10.1016\/0304-3975(76)90086-4","journal-title":"Theor. Comput. Sci."},{"key":"1111_CR7","doi-asserted-by":"crossref","unstructured":"Gronemann, M.: Bitonic st-orderings of biconnected planar graphs. In: Duncan, C.A., Symvonis, A. (eds.) Graph Drawing, vol. 8871 of LNCS, pp. 162\u2013173. Springer, Cham (2014)","DOI":"10.1007\/978-3-662-45803-7_14"},{"key":"1111_CR8","doi-asserted-by":"crossref","unstructured":"Chaplick, S., Chimani, M., Cornelsen, S., Da Lozzo, G., N\u00f6llenburg, M., Patrignani, M., et al.: Planar L-drawings of directed graphs. In: Frati, F., Ma, K. (eds.) Graph Drawing and Network Visualization, vol. 10692 of LNCS, pp. 465\u2013478. Springer, Cham (2017)","DOI":"10.1007\/978-3-319-73915-1_36"},{"issue":"6","key":"1111_CR9","doi-asserted-by":"publisher","first-page":"2527","DOI":"10.1007\/s00453-018-00542-9","volume":"81","author":"P Angelini","year":"2019","unstructured":"Angelini, P., Bekos, M.A., Liotta, G., Montecchiani, F.: Universal slope sets for 1-bend planar drawings. Algorithmica 81(6), 2527\u20132556 (2019). https:\/\/doi.org\/10.1007\/s00453-018-00542-9","journal-title":"Algorithmica"},{"key":"1111_CR10","doi-asserted-by":"crossref","unstructured":"Gronemann, M.: Bitonic st-orderings for upward planar graphs. In: Hu, Y., N\u00f6llenburg, M. (eds.) Graph Drawing and Network Visualization, vol. 9801 of LNCS, pp. 222\u2013235. Springer, Cham (2016)","DOI":"10.1007\/978-3-319-50106-2_18"},{"issue":"5","key":"1111_CR11","doi-asserted-by":"publisher","first-page":"956","DOI":"10.1137\/S0097539794280736","volume":"25","author":"G Di Battista","year":"1996","unstructured":"Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. Comput. 25(5), 956\u2013997 (1996). https:\/\/doi.org\/10.1137\/S0097539794280736","journal-title":"SIAM J. Comput."},{"key":"1111_CR12","unstructured":"Rettner, C.: Flipped Bitonic st-Orderings of Upward Plane Graphs [Bachelor\u2019s Thesis]. University of W\u00fcrzburg (2019)"},{"key":"1111_CR13","doi-asserted-by":"crossref","unstructured":"Bekos, M.A., Di Giacomo, E., Didimo, W., Liotta, G., Montecchiani, F.: Universal slope sets for upward planar drawings. In: Biedl, T.C., Kerren, A. (eds.) Graph Drawing and Network Visualization, vol. 11282 of LNCS, pp. 77\u201391. Springer, Cham (2018)","DOI":"10.1007\/978-3-030-04414-5_6"},{"key":"1111_CR14","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/BF02187705","volume":"1","author":"R Tamassia","year":"1986","unstructured":"Tamassia, R., Tollis, I.G.: A unified approach to visibility representations of planar graphs. Discrete Comput. Geom. 1, 321\u2013341 (1986). https:\/\/doi.org\/10.1007\/BF02187705","journal-title":"Discrete Comput. Geom."},{"key":"1111_CR15","unstructured":"Otten, R.H.J.M., van Wijk, J.G.: Graph representations in interactive layout design. In: Proceedings of the IEEE International Symposium on Circuits and Systems, pp. 914\u2013918 (1978)"},{"issue":"4","key":"1111_CR16","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1007\/BF01961541","volume":"15","author":"G Di Battista","year":"1996","unstructured":"Di Battista, G., Tamassia, R.: On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15(4), 302\u2013318 (1996). https:\/\/doi.org\/10.1007\/BF01961541","journal-title":"Algorithmica"},{"key":"1111_CR17","doi-asserted-by":"crossref","unstructured":"Gutwenger, C., Mutzel, P.: A linear time implementation of SPQR-trees. In: Marks, J. (ed.) Graph Drawing, vol. 1984 of LNCS, pp. 77\u201390. Springer, Cham (2000)","DOI":"10.1007\/3-540-44541-2_8"},{"issue":"8","key":"1111_CR18","doi-asserted-by":"publisher","first-page":"826","DOI":"10.1109\/12.868028","volume":"49","author":"P Bertolazzi","year":"2000","unstructured":"Bertolazzi, P., Di Battista, G., Didimo, W.: Computing orthogonal drawings with the minimum number of bends. IEEE Trans. Comput. 49(8), 826\u2013840 (2000). https:\/\/doi.org\/10.1109\/12.868028","journal-title":"IEEE Trans. Comput."},{"key":"1111_CR19","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/BF02187850","volume":"7","author":"G Di Battista","year":"1992","unstructured":"Di Battista, G., Tamassia, R., Tollis, I.G.: Area requirement and symmetry display of planar upward drawings. Discrete Comput. Geom. 7, 381\u2013401 (1992). https:\/\/doi.org\/10.1007\/BF02187850","journal-title":"Discrete Comput. Geom."},{"issue":"3","key":"1111_CR20","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1142\/S021819590800260X","volume":"18","author":"F Frati","year":"2008","unstructured":"Frati, F.: On minimum area planar upward drawings of directed trees and other families of directed acyclic graphs. Int. J. Comput. Geom. Appl. 18(3), 251\u2013271 (2008). https:\/\/doi.org\/10.1142\/S021819590800260X","journal-title":"Int. J. Comput. Geom. Appl."},{"key":"1111_CR21","doi-asserted-by":"crossref","unstructured":"Lang, S., Murrow, G.: Volumes. In: Geometry\u2014A High School Course, 2nd edn., pp. 261\u2013294. Springer (1997)","DOI":"10.1007\/978-1-4757-2022-8_9"},{"key":"1111_CR22","doi-asserted-by":"crossref","unstructured":"Aigner, M., Ziegler, G.M.: Three applications of Euler\u2019s formula: Pick\u2019s theorem. In Proofs from THE BOOK, 6th edn., pp. 93\u201394. Springer (2018)","DOI":"10.1007\/978-3-662-57265-8_13"},{"key":"1111_CR23","unstructured":"Pick, G.: Geometrisches zur Zahlenlehre. Sonderabdr. Naturw.-medizin. Verein f. B\u00f6hmen \u201cLotos\u201d Nr. 8, 9 S. $$8^{\\circ }$$ (1899)"},{"key":"1111_CR24","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/0304-3975(88)90123-5","volume":"61","author":"G Di Battista","year":"1988","unstructured":"Di Battista, G., Tamassia, R.: Algorithms for plane representations of acyclic digraphs. Theor. Comput. Sci. 61, 175\u2013198 (1988). https:\/\/doi.org\/10.1016\/0304-3975(88)90123-5","journal-title":"Theor. Comput. Sci."}],"container-title":["Algorithmica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00453-023-01111-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00453-023-01111-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00453-023-01111-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,22]],"date-time":"2023-09-22T15:03:30Z","timestamp":1695395010000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00453-023-01111-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,11]]},"references-count":24,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["1111"],"URL":"https:\/\/doi.org\/10.1007\/s00453-023-01111-5","relation":{},"ISSN":["0178-4617","1432-0541"],"issn-type":[{"type":"print","value":"0178-4617"},{"type":"electronic","value":"1432-0541"}],"subject":[],"published":{"date-parts":[[2023,3,11]]},"assertion":[{"value":"1 December 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 February 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 March 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}