{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T16:39:40Z","timestamp":1740155980025,"version":"3.37.3"},"reference-count":44,"publisher":"Wiley","license":[{"start":{"date-parts":[[2024,4,18]],"date-time":"2024-04-18T00:00:00Z","timestamp":1713398400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007613","name":"Majmaah University","doi-asserted-by":"publisher","award":["R-2024-1050"],"award-info":[{"award-number":["R-2024-1050"]}],"id":[{"id":"10.13039\/501100007613","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Operations Research"],"published-print":{"date-parts":[[2024,4,18]]},"abstract":"Let G\u2009=\u2009(V, E) be a connected, basic, and finite graph. A subset T<\/a:mi>=<\/a:mo>u<\/a:mi><\/a:mrow>1<\/a:mn><\/a:mrow><\/a:msub>,<\/a:mo>u<\/a:mi><\/a:mrow>2<\/a:mn><\/a:mrow><\/a:msub>,<\/a:mo>\u2026<\/a:mo>,<\/a:mo>u<\/a:mi><\/a:mrow>k<\/a:mi><\/a:mrow><\/a:msub><\/a:mrow><\/a:mfenced><\/a:math><\/jats:inline-formula> of V(G) is said to be a resolving set if for any y\u2009\u2208\u2009V(G), the code of y with regards to T, represented by C<\/f:mi><\/f:mrow>T<\/f:mi><\/f:mrow><\/f:msub>y<\/f:mi><\/f:mrow><\/f:mfenced><\/f:math><\/jats:inline-formula>, which is defined as C<\/k:mi><\/k:mrow>T<\/k:mi><\/k:mrow><\/k:msub>y<\/k:mi><\/k:mrow><\/k:mfenced>=<\/k:mo>d<\/k:mi>u<\/k:mi><\/k:mrow>1<\/k:mn><\/k:mrow><\/k:msub>,<\/k:mo>y<\/k:mi><\/k:mrow><\/k:mfenced>,<\/k:mo>d<\/k:mi>u<\/k:mi><\/k:mrow>2<\/k:mn><\/k:mrow><\/k:msub>,<\/k:mo>y<\/k:mi><\/k:mrow><\/k:mfenced>,<\/k:mo>\u2026<\/k:mo>,<\/k:mo>d<\/k:mi>u<\/k:mi><\/k:mrow>k<\/k:mi><\/k:mrow><\/k:msub>,<\/k:mo>y<\/k:mi><\/k:mrow><\/k:mfenced><\/k:mrow><\/k:math><\/jats:inline-formula>, is different for various y. The dimension of G is the smallest cardinality of a resolving set and is denoted by dim(G). If, for any t\u2009 \u2208\u2009 V \u2013 S, there exists r\u2009\u2208\u2009S such that S<\/y:mi>\u2013<\/y:mo>r<\/y:mi><\/y:mrow><\/y:mfenced><\/y:mrow><\/y:mrow><\/y:mfenced>\u222a<\/y:mo>t<\/y:mi><\/y:mrow><\/y:mfenced><\/y:math><\/jats:inline-formula> is a resolving set, then the resolving set S is secure. The secure metric dimension of \ud835\udc3a is the cardinal number of the minimum secure resolving set. Determining the secure metric dimension of any given graph is an NP-complete problem. In addition, there are several uses for the metric dimension in a variety of fields, including image processing, pattern recognition, network discovery and verification, geographic routing protocols, and combinatorial optimization. In this paper, we determine the secure metric dimension of special graphs such as a globe graph G<\/jb:mi>l<\/jb:mi><\/jb:mrow>n<\/jb:mi><\/jb:mrow><\/jb:msub><\/jb:math><\/jats:inline-formula>, flag graph F<\/lb:mi>l<\/lb:mi><\/lb:mrow>n<\/lb:mi><\/lb:mrow><\/lb:msub><\/lb:math><\/jats:inline-formula>, H- graph of path P<\/nb:mi><\/nb:mrow>n<\/nb:mi><\/nb:mrow><\/nb:msub><\/nb:math><\/jats:inline-formula>, a bistar graph B<\/pb:mi><\/pb:mrow>n<\/pb:mi>,<\/pb:mo>n<\/pb:mi><\/pb:mrow>2<\/pb:mn><\/pb:mrow><\/pb:msubsup><\/pb:math><\/jats:inline-formula>, and tadpole graph T<\/rb:mi><\/rb:mrow>3<\/rb:mn>,<\/rb:mo>m<\/rb:mi><\/rb:mrow><\/rb:msub><\/rb:math><\/jats:inline-formula>. Finally, we derive the explicit formulas for the secure metric dimension of tadpole graph T<\/tb:mi><\/tb:mrow>n<\/tb:mi>,<\/tb:mo>m<\/tb:mi><\/tb:mrow><\/tb:msub><\/tb:math><\/jats:inline-formula>, subdivision of tadpole graph S<\/vb:mi>T<\/vb:mi><\/vb:mrow>3<\/vb:mn>,<\/vb:mo>m<\/vb:mi><\/vb:mrow><\/vb:msub><\/vb:mrow><\/vb:mfenced><\/vb:math><\/jats:inline-formula>, and subdivision of tadpole graph S<\/ac:mi>T<\/ac:mi><\/ac:mrow>n<\/ac:mi>,<\/ac:mo>m<\/ac:mi><\/ac:mrow><\/ac:msub><\/ac:mrow><\/ac:mfenced><\/ac:math><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1155\/2024\/3084976","type":"journal-article","created":{"date-parts":[[2024,4,18]],"date-time":"2024-04-18T19:35:08Z","timestamp":1713468908000},"page":"1-6","source":"Crossref","is-referenced-by-count":1,"title":["The Secure Metric Dimension of the Globe Graph and the Flag Graph"],"prefix":"10.1155","volume":"2024","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2050-5236","authenticated-orcid":true,"given":"Sultan","family":"Almotairi","sequence":"first","affiliation":[{"name":"Department of Computer Science, Faculty of College of Computer and Information Sciences, Majmaah University, Al Majmaah 11952, Saudi Arabia"}]},{"given":"Olayan","family":"Alharbi","sequence":"additional","affiliation":[{"name":"Department of Information Systems, College of Computer and Information Sciences, Majmaah University, Al Majmaah 11952, Saudi Arabia"}]},{"given":"Zaid","family":"Alzaid","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Faculty of Computer and Information Systems, Islamic University of Madinah, Madinah 42351, Saudi Arabia"}]},{"given":"Badr","family":"Almutairi","sequence":"additional","affiliation":[{"name":"Department of Information Technology, College of Computer Sciences and Information Technology College, Majmaah University, Al Majmaah 11952, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7009-6912","authenticated-orcid":true,"given":"Basma","family":"Mohamed","sequence":"additional","affiliation":[{"name":"Mathematics and Computer Science Department, Faculty of Science, Menoufia University, Shibin El Kom 32511, Egypt"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/s0166-218x(00)00198-0"},{"first-page":"549","article-title":"Leaves of trees","author":"P. 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