{"id":"https://openalex.org/W2067239519","doi":"https://doi.org/10.1002/net.20406","title":"Optimum steiner ratio for gradient-constrained networks connecting three points in 3-space, part II: The gradient-constraint m satisfies \\documentclass{article} \\usepackage{amsmath,amsfonts,amssymb}\\pagestyle{empty}\\begin{document}$ 1 \\leq m \\leq \\sqrt{3}","display_name":"Optimum steiner ratio for gradient-constrained networks connecting three points in 3-space, part II: The gradient-constraint m satisfies \\documentclass{article} \\usepackage{amsmath,amsfonts,amssymb}\\pagestyle{empty}\\begin{document}$ 1 \\leq m \\leq \\sqrt{3}","publication_year":2010,"publication_date":"2010-10-01","ids":{"openalex":"https://openalex.org/W2067239519","doi":"https://doi.org/10.1002/net.20406","mag":"2067239519"},"language":"en","primary_location":{"id":"doi:10.1002/net.20406","doi":"https://doi.org/10.1002/net.20406","is_oa":false,"landing_page_url":"https://doi.org/10.1002/net.20406","pdf_url":null,"source":{"id":"https://openalex.org/S191798613","display_name":"Networks","issn_l":"0028-3045","issn":["0028-3045","1097-0037"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","ccf-c","cwts-core","dongbi-b","jufo-2","norway-1","poland-140","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Networks","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5065822881","display_name":"Doreen Anne Thomas","orcid":"https://orcid.org/0000-0002-8141-3362","observed_orcids":["https://orcid.org/0000-0002-8141-3362"]},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"The University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":true,"raw_author_name":"Doreen Thomas","raw_affiliation_strings":["Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia","Department of Mechanical Engineering; The University of Melbourne; Victoria 3010 Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia","institution_ids":["https://openalex.org/I165779595"]},{"raw_affiliation_string":"Department of Mechanical Engineering; The University of Melbourne; Victoria 3010 Australia","institution_ids":["https://openalex.org/I165779595"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5081620539","display_name":"Kevin Prendergast","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"The University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Kevin Prendergast","raw_affiliation_strings":["Department of Electrical and Electronic Engineering, The University of Melbourne, Victoria 3010, Australia","Department of Electrical & Electronic Engineering The University of Melbourne  Victoria 3010 Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Electronic Engineering, The University of Melbourne, Victoria 3010, Australia","institution_ids":["https://openalex.org/I165779595"]},{"raw_affiliation_string":"Department of Electrical & Electronic Engineering The University of Melbourne  Victoria 3010 Australia","institution_ids":["https://openalex.org/I165779595"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5054154444","display_name":"Jia Feng Weng","orcid":"https://orcid.org/0000-0002-6976-2375","observed_orcids":["https://orcid.org/0000-0002-6976-2375"]},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"The University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Jia Weng","raw_affiliation_strings":["Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia","Department of Mechanical Engineering; The University of Melbourne; Victoria 3010 Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia","institution_ids":["https://openalex.org/I165779595"]},{"raw_affiliation_string":"Department of Mechanical Engineering; The University of Melbourne; Victoria 3010 Australia","institution_ids":["https://openalex.org/I165779595"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5065822881"],"corresponding_institution_ids":["https://openalex.org/I165779595"],"apc_list":{"value":3750,"currency":"USD","value_usd":3750},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.2133732,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"57","issue":"4","first_page":"354","last_page":"361"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9824000000953674,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9824000000953674,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.01140000019222498,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.002400000113993883,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/steiner-ratio","display_name":"Steiner ratio","score":0.9829999804496765},{"id":"https://openalex.org/keywords/steiner-minimal-tree","display_name":"minimum Steiner tree","score":0.8740000128746033},{"id":"https://openalex.org/keywords/minimum-spanning-tree","display_name":"minimum spanning tree","score":0.8180000185966492},{"id":"https://openalex.org/keywords/steiner-points","display_name":"Steiner points","score":0.546999990940094},{"id":"https://openalex.org/keywords/network-optimization","display_name":"network optimization","score":0.48100000619888306},{"id":"https://openalex.org/keywords/three-dimensional-space","display_name":"three-dimensional space","score":0.5329999923706055}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7282308340072632},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5068293213844299},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.4765746593475342},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.47166740894317627},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.4701417088508606},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.37880194187164307},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16527393460273743},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15921831130981445},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1002/net.20406","doi":"https://doi.org/10.1002/net.20406","is_oa":false,"landing_page_url":"https://doi.org/10.1002/net.20406","pdf_url":null,"source":{"id":"https://openalex.org/S191798613","display_name":"Networks","issn_l":"0028-3045","issn":["0028-3045","1097-0037"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","ccf-c","cwts-core","dongbi-b","jufo-2","norway-1","poland-140","russia-white-list-2","vabb-shw"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Networks","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W1600385162","https://openalex.org/W1964421131","https://openalex.org/W2024833542","https://openalex.org/W2056957755","https://openalex.org/W2070817156","https://openalex.org/W2075207773","https://openalex.org/W4241715968"],"related_works":["https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W2072657027","https://openalex.org/W2600246793","https://openalex.org/W4238204885"],"abstract_inverted_index":{"The":[0],"authors":[1],"have":[2],"proved":[3],"in":[4,51],"a":[5,19,27,32],"previous":[6],"article":[7],"that":[8,72],"the":[9,16,24,44,52,67,78,91],"Steiner":[10,20],"ratio":[11,14,92],"(the":[12],"minimum":[13],"of":[15,18,26,48,81],"length":[17,25],"minimal":[21,28],"tree":[22],"to":[23,70],"spanning":[29],"tree)":[30],"for":[31],"gradient-constrained":[33],"network":[34,53],"on":[35],"three":[36],"points":[37],"is":[38],"\\documentclass{article}":[39,84,94,99],"\\usepackage{amsmath,amsfonts,amssymb}\\pagestyle{empty}\\begin{document}$":[40,85,95,100],"1/2+(\\sqrt{4+3m^2}-1)/(4\\sqrt{1+m^2})":[41],"$\\end{document}":[42,97],"if":[43,98],"maximum":[45,79],"gradient":[46],"m":[47,55,82],"an":[49],"edge":[50],"satisfies":[54,83],"\u2264":[56],"1.":[57],"In":[58],"this":[59,63,73],"article,":[60],"we":[61],"continue":[62],"study":[64],"and":[65],"use":[66],"same":[68],"strategy":[69],"show":[71],"result":[74],"still":[75],"holds":[76],"when":[77],"value":[80],"1\\le":[86],"m\\le":[87],"\\sqrt{3}":[88,102],"$\\end{document}.":[89,103],"However":[90],"becomes":[93],"(3m+\\sqrt{9m^{2}+12})/(4\\sqrt{3m^{2}+3})":[96],"m\\ge":[101],"\u00a9":[104],"2010":[105],"Wiley":[106],"Periodicals,":[107],"Inc.":[108],"NETWORKS,":[109],"Vol.":[110],"57(4),":[111],"354\u2013361":[112]},"counts_by_year":[],"updated_date":"2026-10-12T09:40:04.860316","created_date":"2025-10-10T00:00:00"}