{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,19]],"date-time":"2024-09-19T16:13:53Z","timestamp":1726762433247},"reference-count":39,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,16]],"date-time":"2021-07-16T00:00:00Z","timestamp":1626393600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007801","name":"Fundaci\u00f3n S\u00e9neca","doi-asserted-by":"publisher","award":["20783\/PI\/18"],"id":[{"id":"10.13039\/100007801","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PGC2018-097198-B-I00"],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Fuzzy graphs (FGs), broadly known as fuzzy incidence graphs (FIGs), are an applicable and well-organized tool to epitomize and resolve multiple real-world problems in which ambiguous data and information are essential. In this article, we extend the idea of domination of FGs to the FIG using strong pairs. An idea of strong pair dominating set and a strong pair domination number (SPDN) is explained with various examples. A theorem to compute SPDN for a complete fuzzy incidence graph (CFIG) is also provided. It is also proved that in any fuzzy incidence cycle (FIC) with l vertices the minimum number of elements in a strong pair dominating set are M[\u03b3s(Cl(\u03c3,\u03d5,\u03b7))]=\u2308l3\u2309. We define the joining of two FIGs and present a way to compute SPDN in the join of FIGs. A theorem to calculate SPDN in the joining of two strong fuzzy incidence graphs is also provided. An innovative idea of accurate domination of FIGs is also proposed. Some instrumental and useful results of accurate domination for FIC are also obtained. In the end, a real-life application of SPDN to find which country\/countries has\/have the best trade policies among different countries is examined. Our proposed method is symmetrical to the optimization.<\/jats:p>","DOI":"10.3390\/sym13071279","type":"journal-article","created":{"date-parts":[[2021,7,19]],"date-time":"2021-07-19T01:18:52Z","timestamp":1626657532000},"page":"1279","source":"Crossref","is-referenced-by-count":11,"title":["Domination in Join of Fuzzy Incidence Graphs Using Strong Pairs with Application in Trading System of Different Countries"],"prefix":"10.3390","volume":"13","author":[{"given":"Irfan","family":"Nazeer","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-8691-1088","authenticated-orcid":false,"given":"Tabasam","family":"Rashid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}]},{"given":"Muhammad Tanveer","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan"}]},{"ORCID":"http:\/\/orcid.org\/0000-0003-2788-809X","authenticated-orcid":false,"given":"Juan Luis Garc\u00eda","family":"Guirao","sequence":"additional","affiliation":[{"name":"Departamento de Matematica Aplicada y Estadistica, Universidad Politecnica de Cartagena, Hospital de Marina, 30203 Cartagena, Spain"},{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)\u2014Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. 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Today Proc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1279\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,15]],"date-time":"2024-07-15T14:41:46Z","timestamp":1721054506000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1279"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,16]]},"references-count":39,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["sym13071279"],"URL":"https:\/\/doi.org\/10.3390\/sym13071279","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,16]]}}}  <script>parent.window.postMessage('ixistenz.url:https://api.crossref.org/works/10.3390%2FSYM13071279', '*'); </script> <div id='browserdiffcontainer' style='position: fixed; right: 10px; top: 10vh; min-width: 60px;  border: 1px solid white; background: rgb(251, 189, 0);; opacity: 0.9; font-family: helvetica, arial, sans serif;  font-size: 12px; z-index: 20001; padding-bottom: 20px;'><div onclick="toggle('browserdiffcontainerlist'); return false;" style='background: rgb(151, 89, 0);; color: white; '><strong>&nbsp; <span class='glyphicon glyphicon-minus'></span> NODES</strong></div><div id='browserdiffcontainerlist' style='padding-left: 10px; padding-right: 5px; max-height: 50vh; overflow: auto; '><div><a href='#diffid884' style='color: white;' onClick="parent.window.postMessage('ixistenz.opendiff:884', '*');; return false;" style='text-decoration: none; '>Idea </a> <a style='color: white;' href='#diffid884'><sup>3</sup></a></div><div><a href='#diffid119' style='color: white;' onClick="parent.window.postMessage('ixistenz.opendiff:119', '*');; return false;" style='text-decoration: none; '>idea </a> <a style='color: white;' href='#diffid119'><sup>3</sup></a></div><div><a href='#diffid878' style='color: white;' onClick="parent.window.postMessage('ixistenz.opendiff:878', '*');; return false;" style='text-decoration: none; '>Note </a> <a style='color: white;' href='#diffid878'><sup>1</sup></a></div></div></div>
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