{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:04:54Z","timestamp":1740107094760,"version":"3.37.3"},"reference-count":9,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2020,6,12]],"date-time":"2020-06-12T00:00:00Z","timestamp":1591920000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,6,12]],"date-time":"2020-06-12T00:00:00Z","timestamp":1591920000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2020,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Given a set of sources and a set of sinks as points in the Euclidean plane, a<jats:italic>directed network<\/jats:italic>is a directed graph drawn in the plane with a directed path from each source to each sink. Such a network may contain nodes other than the given sources and sinks, called Steiner points. We characterize the local structure of the Steiner points in all shortest-length directed networks in the Euclidean plane. This characterization implies that these networks are constructible by straightedge and compass. Our results build on unpublished work of Alfaro, Campbell, Sher, and Soto from 1989 and 1990. Part of the proof is based on a new method that uses other norms in the plane. This approach gives more conceptual proofs of some of their results, and as a consequence, we also obtain results on shortest directed networks for these norms.<\/jats:p>","DOI":"10.1007\/s00373-020-02183-8","type":"journal-article","created":{"date-parts":[[2020,6,12]],"date-time":"2020-06-12T20:02:16Z","timestamp":1591992136000},"page":"1457-1475","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Shortest Directed Networks in the Plane"],"prefix":"10.1007","volume":"36","author":[{"given":"Alastair","family":"Maxwell","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1668-887X","authenticated-orcid":false,"given":"Konrad J.","family":"Swanepoel","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,6,12]]},"reference":[{"key":"2183_CR1","unstructured":"Alfaro, M.: Existence of shortest directed networks in $${\\mathbf{R}}^2$$. Technical report, Williams College (1990)"},{"key":"2183_CR2","doi-asserted-by":"publisher","first-page":"201","DOI":"10.2140\/pjm.1995.167.201","volume":"167","author":"M Alfaro","year":"1995","unstructured":"Alfaro, M.: Existence of shortest directed networks in $${\\mathbf{R}}^2$$. Pac. J. Math. 167, 201\u2013214 (1995)","journal-title":"Pac. J. Math."},{"key":"2183_CR3","unstructured":"Alfaro, M., Campbell, T., Sher, J., Soto, A.: Length minimizing directed networks can meet in fours. Technical report, Williams College (1989)"},{"key":"2183_CR4","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s00407-013-0127-z","volume":"68","author":"M Brazil","year":"2014","unstructured":"Brazil, M., Graham, R.L., Thomas, D.A., Zachariasen, M.: On the history of the Euclidean Steiner tree problem. Arch. Hist. Exact Sci. 68, 327\u2013354 (2014)","journal-title":"Arch. Hist. Exact Sci."},{"key":"2183_CR5","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-13915-9","volume-title":"Optimal Interconnection Trees in the Plane. Algorithms and Combinatorics","author":"M Brazil","year":"2015","unstructured":"Brazil, M., Zachariasen, M.: Optimal Interconnection Trees in the Plane. Algorithms and Combinatorics, vol. 29. Springer, Cham (2015)"},{"key":"2183_CR6","volume-title":"100 Great Problems of Elementary Mathematics. Reprint of the 1965 Edition","author":"H D\u00f6rrie","year":"1982","unstructured":"D\u00f6rrie, H.: 100 Great Problems of Elementary Mathematics. Reprint of the 1965 Edition. Dover, New York (1982)"},{"key":"2183_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/0116001","volume":"16","author":"EN Gilbert","year":"1968","unstructured":"Gilbert, E.N., Pollak, H.O.: Steiner minimal trees. SIAM J. Appl. Math. 16, 1\u201329 (1968)","journal-title":"SIAM J. Appl. 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