{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T02:08:35Z","timestamp":1740103715076,"version":"3.37.3"},"reference-count":15,"publisher":"Wiley","license":[{"start":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T00:00:00Z","timestamp":1600905600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100015673","name":"Anhui Jianzhu University","doi-asserted-by":"publisher","award":["2016QD116","2017dc03"],"award-info":[{"award-number":["2016QD116","2017dc03"]}],"id":[{"id":"10.13039\/501100015673","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100015673","name":"Anhui Jianzhu University","doi-asserted-by":"publisher","award":["2016QD116","2017dc03"],"award-info":[{"award-number":["2016QD116","2017dc03"]}],"id":[{"id":"10.13039\/501100015673","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2020,9,24]]},"abstract":"<jats:p>Let <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>G<\/mml:mi><\/mml:math> be a finite, connected graph of order of, at least, 2 with vertex set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>V<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math> and edge set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mi>E<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>. A set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>S<\/mml:mi><\/mml:math> of vertices of the graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi>G<\/mml:mi><\/mml:math> is a doubly resolving set for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi>G<\/mml:mi><\/mml:math> if every two distinct vertices of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>G<\/mml:mi><\/mml:math> are doubly resolved by some two vertices of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mi>S<\/mml:mi><\/mml:math>. The minimal doubly resolving set of vertices of graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mi>G<\/mml:mi><\/mml:math> is a doubly resolving set with minimum cardinality and is denoted by <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mi>\u03c8<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>. In this paper, first, we construct a class of graphs of order <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mn>2<\/mml:mn><mml:mi>n<\/mml:mi><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>\u03a3<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>r<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><mml:mo>\u2212<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:msup><mml:mrow><mml:mtext>nm<\/mml:mtext><\/mml:mrow><mml:mrow><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math>, denoted by <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mtext>LSG<\/mml:mtext><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>m<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>, and call these graphs as the layer Sun graphs with parameters <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mi>n<\/mml:mi><\/mml:math>, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mi>m<\/mml:mi><\/mml:math>, and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mi>k<\/mml:mi><\/mml:math>. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of the layer Sun graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mtext>LSG<\/mml:mtext><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>m<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math> and the line graph of the layer Sun graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mtext>LSG<\/mml:mtext><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>m<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2020\/6267072","type":"journal-article","created":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T23:32:55Z","timestamp":1600990375000},"page":"1-8","source":"Crossref","is-referenced-by-count":7,"title":["Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of the Layer Sun Graph"],"prefix":"10.1155","volume":"2020","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9620-7692","authenticated-orcid":true,"given":"Jia-Bao","family":"Liu","sequence":"first","affiliation":[{"name":"School of Finance and Mathematics, Huainan Normal University, Huainan 232038, China"},{"name":"School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7942-5356","authenticated-orcid":true,"given":"Ali","family":"Zafari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Payame Noor University, P.O. 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