{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T08:16:49Z","timestamp":1763367409639,"version":"3.40.5"},"reference-count":26,"publisher":"Wiley","license":[{"start":{"date-parts":[[2020,5,29]],"date-time":"2020-05-29T00:00:00Z","timestamp":1590710400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Project of Anhui Jianzhu University","award":["2016QD116","2017dc03"],"award-info":[{"award-number":["2016QD116","2017dc03"]}]},{"name":"Project of Anhui Jianzhu University","award":["2016QD116","2017dc03"],"award-info":[{"award-number":["2016QD116","2017dc03"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2020,5,29]]},"abstract":"<jats:p>Let <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi mathvariant=\"normal\">\u0393<\/mml:mi><\/mml:math> be a simple connected undirected graph 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 mathvariant=\"normal\">\u0393<\/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 mathvariant=\"normal\">\u0393<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math>. The metric dimension of a graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi mathvariant=\"normal\">\u0393<\/mml:mi><\/mml:math> is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. For an ordered subset <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi>W<\/mml:mi><mml:mo>=<\/mml:mo><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><\/mml:math> of vertices in a graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi mathvariant=\"normal\">\u0393<\/mml:mi><\/mml:math> and a vertex <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>v<\/mml:mi><\/mml:math> of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mi mathvariant=\"normal\">\u0393<\/mml:mi><\/mml:math>, the metric representation of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mi>v<\/mml:mi><\/mml:math> with respect to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mi>W<\/mml:mi><\/mml:math> is the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mi>k<\/mml:mi><\/mml:math>-vector <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mi>r<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mfenced open=\"\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mi>W<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>d<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>v<\/mml:mi><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><mml:mo>,<\/mml:mo><mml:mi>d<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>v<\/mml:mi><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>d<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>v<\/mml:mi><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>w<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><\/mml:math>. If every pair of distinct vertices of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mi mathvariant=\"normal\">\u0393<\/mml:mi><\/mml:math> have different metric representations, then the ordered set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mi>W<\/mml:mi><\/mml:math> is called a resolving set of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mi mathvariant=\"normal\">\u0393<\/mml:mi><\/mml:math>. It is known that the problem of computing this invariant is NP-hard. In this paper, we consider the problem of determining the cardinality <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mi>\u03c8<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi mathvariant=\"normal\">\u0393<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math> of minimal doubly resolving sets of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mi mathvariant=\"normal\">\u0393<\/mml:mi><\/mml:math> and the strong metric dimension for the jellyfish graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\"><mml:mtext>JFG<\/mml:mtext><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>n<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math> and the cocktail party graph <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M19\"><mml:mtext>CP<\/mml:mtext><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>k<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:mfenced><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2020\/9407456","type":"journal-article","created":{"date-parts":[[2020,5,30]],"date-time":"2020-05-30T00:24:45Z","timestamp":1590798285000},"page":"1-7","source":"Crossref","is-referenced-by-count":17,"title":["Metric Dimension, Minimal Doubly Resolving Sets, and the Strong Metric Dimension for Jellyfish Graph and Cocktail Party 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 Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9758-8821","authenticated-orcid":true,"given":"Ali","family":"Zafari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Payame Noor University, P.O. Box 19395-4697, Tehran, Iran"}]},{"given":"Hassan","family":"Zarei","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Payame Noor University, P.O. Box 19395-4697, Tehran, Iran"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/s0166-218x(00)00198-0"},{"year":"1953","key":"2"},{"key":"3","first-page":"191","volume":"2","year":"1976","journal-title":"Combinatoria"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/bdq096"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1023\/a:1025745406160"},{"key":"7","first-page":"349","volume":"88","year":"2008","journal-title":"Ars Combinatoria"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2011.09.008"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2019.01.002"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.4153\/cmb-2016-048-1"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1109\/jsac.2006.884015"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218x(95)00106-2"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1030.0070"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1137\/050641867"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2013.772999"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.2298\/aadm111116023k"},{"issue":"70","key":"17","first-page":"123","volume":"20","year":"2018","journal-title":"Mathematical Reports"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2006.06.009"},{"key":"19","first-page":"59","volume":"76","year":"2011","journal-title":"Journal of Combinatorial Mathematics and Combinatorial Computing"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/1008\/1\/012032"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1016\/j.cor.2008.08.002"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.5614\/ejgta.2019.7.1.7"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2012.07.025"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1109\/access.2019.2938579"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2013.01.021"},{"year":"1993","key":"26"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.5614\/ejgta.2017.5.1.7"}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2020\/9407456.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2020\/9407456.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2020\/9407456.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,30]],"date-time":"2020-05-30T00:24:52Z","timestamp":1590798292000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/complexity\/2020\/9407456\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,29]]},"references-count":26,"alternative-id":["9407456","9407456"],"URL":"https:\/\/doi.org\/10.1155\/2020\/9407456","relation":{},"ISSN":["1076-2787","1099-0526"],"issn-type":[{"type":"print","value":"1076-2787"},{"type":"electronic","value":"1099-0526"}],"subject":[],"published":{"date-parts":[[2020,5,29]]}}}