{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T03:11:42Z","timestamp":1766113902157,"version":"3.48.0"},"reference-count":52,"publisher":"Informa UK Limited","issue":"3","license":[{"start":{"date-parts":[[2025,4,22]],"date-time":"2025-04-22T00:00:00Z","timestamp":1745280000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007414","name":"Qassim University","doi-asserted-by":"publisher","award":["QU-APC-2025"],"award-info":[{"award-number":["QU-APC-2025"]}],"id":[{"id":"10.13039\/501100007414","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.tandfonline.com"],"crossmark-restriction":true},"short-container-title":["AKCE International Journal of Graphs and Combinatorics"],"published-print":{"date-parts":[[2025,9,2]]},"DOI":"10.1080\/09728600.2025.2487427","type":"journal-article","created":{"date-parts":[[2025,4,22]],"date-time":"2025-04-22T17:59:09Z","timestamp":1745344749000},"page":"253-264","update-policy":"https:\/\/doi.org\/10.1080\/tandf_crossmark_01","source":"Crossref","is-referenced-by-count":0,"title":["Investigating the Metric and Fault-Tolerant Dimensions in Para-Line Network Topologies"],"prefix":"10.1080","volume":"22","author":[{"given":"M.","family":"Faheem","sequence":"first","affiliation":[{"name":"Department of Mathematics, School of Science, University of Management and Technology","place":["Lahore, Pakistan"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Science, University of Management and Technology","place":["Lahore, Pakistan"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Z.","family":"Zahid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Science, University of Management and Technology","place":["Lahore, Pakistan"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Njood. S. E.","family":"ALmutire","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University","place":["Buraydah, Saudi Arabia"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hamiden Abd El-Wahed","family":"Khalifa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University","place":["Buraydah, Saudi Arabia"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1466-8821","authenticated-orcid":false,"given":"Taha","family":"Radwan","sequence":"additional","affiliation":[{"name":"Department of Management Information Systems, College of Business and Economics, Qassim University","place":["Buraydah, Saudi Arabia"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"301","published-online":{"date-parts":[[2025,4,22]]},"reference":[{"key":"e_1_3_5_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/CyberC.2018.00072"},{"key":"e_1_3_5_3_1","doi-asserted-by":"publisher","DOI":"10.3390\/computation11100188"},{"key":"e_1_3_5_4_1","doi-asserted-by":"publisher","DOI":"10.1051\/matecconf\/202237204005"},{"key":"e_1_3_5_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(00)00126-7"},{"key":"e_1_3_5_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(95)00106-2"},{"key":"e_1_3_5_7_1","doi-asserted-by":"publisher","DOI":"10.1109\/JSAC.2013.SUP2.12130014"},{"key":"e_1_3_5_8_1","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-matsci-070909-104532"},{"key":"e_1_3_5_9_1","first-page":"549","article-title":"Leaves of Trees","volume":"14","author":"Slater P. J.","year":"1975","unstructured":"Slater, P. J. Leaves of Trees. In: Proceeding of the 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing, Congressus Numerantium, 1975, 14, 549\u2013559.","journal-title":"Proceeding of the 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing, Congressus Numerantium"},{"key":"e_1_3_5_10_1","first-page":"445","article-title":"Dominating and Reference Set in a Graph","volume":"22","author":"Slater P. J.","year":"1988","unstructured":"Slater, P. J. Dominating and Reference Set in a Graph. J. Math. Phys. Sci. 1988, 22, 445\u2013455.","journal-title":"J. Math. Phys. Sci"},{"key":"e_1_3_5_11_1","first-page":"191","article-title":"On the metric dimension of a graph","volume":"2","author":"Harary F.","year":"1976","unstructured":"Harary, F.; Melter, R. A. On the metric dimension of a graph. Ars Comb. 1976, 2, 191\u2013195.","journal-title":"Ars Comb"},{"key":"e_1_3_5_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(00)00198-0"},{"key":"e_1_3_5_13_1","first-page":"47","article-title":"The Theory and Applications of Resolvability in Graphs: A Survey","volume":"160","author":"Chartrand G.","year":"2003","unstructured":"Chartrand, G.; Zhang, P. The Theory and Applications of Resolvability in Graphs: A Survey. Congr. Numer. 2003, 160, 47\u201368.","journal-title":"Congr. Numer"},{"key":"e_1_3_5_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2014.09.045"},{"key":"e_1_3_5_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2017.07.027"},{"volume-title":"Computers and Intractability: A Guide to the Theory of NP-Completeness","author":"Garey M. R.","key":"e_1_3_5_16_1","unstructured":"Garey, M. R.; Johnson, D. S. Computers and Intractability: A Guide to the Theory of NP-Completeness. New York, NY: W. H. Freeman."},{"key":"e_1_3_5_17_1","first-page":"273","article-title":"On the Metric Dimension of Some Families of Graphs","volume":"29","author":"Caceres J.","year":"2007","unstructured":"Caceres, J.; Hernando, C.; Mora, M.; Pelayo, I. M.; Puertas, M. L.; Seara, C.; Wood, D. R. On the Metric Dimension of Some Families of Graphs. Electron. Notes Discrete Math. 2007, 29, 273\u2013277.","journal-title":"Electron. Notes Discrete Math"},{"key":"e_1_3_5_18_1","first-page":"103","article-title":"On the Metric Dimension of a Graph","volume":"137","author":"Fouquet J. L.","year":"1994","unstructured":"Fouquet, J. L.; Jolivet, J. C.; Meyniel, H. On the Metric Dimension of a Graph. Discrete Math. 1994, 137, 103\u2013116.","journal-title":"Discrete Math"},{"key":"e_1_3_5_19_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2005.09.015"},{"key":"e_1_3_5_20_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jda.2006.09.002"},{"key":"e_1_3_5_21_1","first-page":"81","article-title":"Fault-Tolerant Metric Dimension of Graph","volume":"5","author":"Hernando C.","year":"2008","unstructured":"Hernando, C.; Mora, M.; Slater, P. J.; Wood, D. R. Fault-Tolerant Metric Dimension of Graph. Proc. Int. Conf. Convexity Discrete Struct. Ramanujan Math. Soc. Lecture Notes 2008, 5, 81\u201385.","journal-title":"Proc. Int. Conf. Convexity Discrete Struct. Ramanujan Math. Soc. Lecture Notes"},{"key":"e_1_3_5_22_1","first-page":"1455","article-title":"Fault-Tolerant Metric Dimension of Lexicographic Product Graphs","volume":"31","author":"Estrada-Moreno A.","year":"2016","unstructured":"Estrada-Moreno, A.; Oellermann, O. R.; Ramirez-Cruz, Y. Fault-Tolerant Metric Dimension of Lexicographic Product Graphs. J. Comb. Optim. 2016, 31, 1455\u20131477.","journal-title":"J. Comb. Optim"},{"key":"e_1_3_5_23_1","doi-asserted-by":"publisher","DOI":"10.1007\/s12190-018-01225-y"},{"key":"e_1_3_5_24_1","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2020.3014883"},{"key":"e_1_3_5_25_1","first-page":"1","article-title":"Fault-Tolerant Basis of Generalized Fat Trees and Perfect Binary Tree Derived Architectures","author":"Prabhu S.","year":"2024","unstructured":"Prabhu, S.; Manimozhi, V.; Davoodi, A.; Guirao, J. L. G. Fault-Tolerant Basis of Generalized Fat Trees and Perfect Binary Tree Derived Architectures. J. Supercomput. 2024; pp 1\u201316.","journal-title":"J. Supercomput."},{"key":"e_1_3_5_26_1","article-title":"Fault-Tolerant Resolvability in Some Classes of Line Graphs","volume":"8","author":"Guo X.","year":"2020","unstructured":"Guo, X.; Faheem, M.; Zahid, Z.; Nazeer, W.; Li, J. (2020). Fault-Tolerant Resolvability in Some Classes of Line Graphs. Math. Probl. Eng. 2020, 8.","journal-title":"Math. Probl. Eng"},{"key":"e_1_3_5_27_1","doi-asserted-by":"publisher","DOI":"10.1155\/2022\/5884924"},{"key":"e_1_3_5_28_1","first-page":"126897","article-title":"Twin Vertices in Fault-Tolerant Metric Sets and Fault-Tolerant Metric Dimension of Multistage Interconnection Networks","volume":"420","author":"Prabhu S.","year":"2022","unstructured":"Prabhu, S.; Manimozhi, V.; Arulperumjothi, M.; Klavzar, S. Twin Vertices in Fault-Tolerant Metric Sets and Fault-Tolerant Metric Dimension of Multistage Interconnection Networks. Appl. Math. Comput. 2022, 420, 126897.","journal-title":"Appl. Math. Comput"},{"key":"e_1_3_5_29_1","doi-asserted-by":"publisher","DOI":"10.1155\/2022\/7688702"},{"key":"e_1_3_5_30_1","doi-asserted-by":"publisher","DOI":"10.1109\/ICDSAAI59313.2023.10452547"},{"key":"e_1_3_5_31_1","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/1714\/1\/012029"},{"key":"e_1_3_5_32_1","doi-asserted-by":"publisher","DOI":"10.1155\/2022\/5784800"},{"key":"e_1_3_5_33_1","first-page":"172","article-title":"On the Fault-Tolerant Metric Dimension of Convex Polytopes","volume":"339","author":"Raza H.","year":"2018","unstructured":"Raza, H.; Hayat, S.; Pan, X. F. On the Fault-Tolerant Metric Dimension of Convex Polytopes. Appl. Math. Comput. 2018, 339, 172\u2013185.","journal-title":"Appl. Math. Comput"},{"key":"e_1_3_5_34_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2019.08.032"},{"key":"e_1_3_5_35_1","doi-asserted-by":"publisher","DOI":"10.1109\/TSMC.2023.3327432"},{"key":"e_1_3_5_36_1","doi-asserted-by":"publisher","DOI":"10.3390\/s151024735"},{"key":"e_1_3_5_37_1","doi-asserted-by":"publisher","DOI":"10.1109\/SRDS.2010.13"},{"key":"e_1_3_5_38_1","first-page":"2450002","article-title":"On Fault-Tolerant Metric Dimension of Supramolecular Networks","volume":"2024","author":"Siddiqui H. M. A.","year":"2024","unstructured":"Siddiqui, H. M. A.; Mazhar, K.; Siddiqui, M. K.; Nadeem, M. F. (2024). On Fault-Tolerant Metric Dimension of Supramolecular Networks. Disc. Math. Algorithms Appl. 2024, 2024, 2450002.","journal-title":"Disc. Math. Algorithms Appl."},{"key":"e_1_3_5_39_1","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2024.3524408"},{"key":"e_1_3_5_40_1","article-title":"Edge Metric Dimension of Some Classes of Toeplitz Networks","volume":"3402275","author":"Alrowaili D.","year":"2021","unstructured":"Alrowaili, D.; Zahid, Z.; Ahsan, M.; Zafar, S.; Siddique, I. Edge Metric Dimension of Some Classes of Toeplitz Networks. J. Math. 2021, 2021, 3402275.","journal-title":"J. Math"},{"key":"e_1_3_5_41_1","first-page":"17","article-title":"The Metric Dimension of Unicyclic Graphs","volume":"40","author":"Poisson C.","year":"2002","unstructured":"Poisson, C.; Zhang, P. The Metric Dimension of Unicyclic Graphs. J. Combin. Math. Combin. Comput. 2002, 40, 17\u201332.","journal-title":"J. Combin. Math. Combin. Comput"},{"key":"e_1_3_5_42_1","first-page":"263","article-title":"Fault-Tolerance in Resolvability","volume":"80","author":"Javaid I.","year":"2009","unstructured":"Javaid, I.; Salman, M.; Chaudhry, M. A.; Shokat, S. Fault-Tolerance in Resolvability. Utilitas Math. 2009, 80, 263\u2013275.","journal-title":"Utilitas Math"},{"key":"e_1_3_5_43_1","first-page":"260","article-title":"Topological Properties of Para-Line Graphs for Chemical Networks","volume":"42","author":"Rashid M. A.","year":"2020","unstructured":"Rashid, M. A.; Ahmad, S.; Siddiqui, M. K.; Imran, M. Topological Properties of Para-Line Graphs for Chemical Networks. PACs 2020, 42, 260\u2013276.","journal-title":"PACs"},{"key":"e_1_3_5_44_1","first-page":"216","article-title":"On Eccentricity-based Topological Indices of Line and Para-Line Graphs of Some Convex Polytopes","volume":"7","author":"Turac T.","year":"2023","unstructured":"Turac, T.; Durgut, R. (2023). On Eccentricity-based Topological Indices of Line and Para-Line Graphs of Some Convex Polytopes. J. Inform. Optim. Sci. 2023, 7, 216\u2013225.","journal-title":"J. Inform. Optim. Sci."},{"key":"e_1_3_5_45_1","doi-asserted-by":"publisher","DOI":"10.33263\/BRIAC113.99159927"},{"key":"e_1_3_5_46_1","doi-asserted-by":"publisher","DOI":"10.1155\/2021\/7057412"},{"key":"e_1_3_5_47_1","doi-asserted-by":"publisher","DOI":"10.1155\/2021\/4611199"},{"key":"e_1_3_5_48_1","first-page":"2829","article-title":"The k\u2212 metric dimension of a graph","volume":"9","author":"Estrado-Moreno A.","year":"2015","unstructured":"Estrado-Moreno, A.; Rodriguez-Velaquez, J. A.; Yero, I. G. The k\u2212 metric dimension of a graph. AMIS. 2015, 9, 2829\u20132840.","journal-title":"AMIS"},{"key":"e_1_3_5_49_1","first-page":"143","article-title":"Edge Metric Dimension of Graphs","volume":"147","author":"Nasir R.","year":"2019","unstructured":"Nasir, R.; Zafar, S.; Zahid, Z. Edge Metric Dimension of Graphs. Ars Combin. 2019, 147, 143\u2013155.","journal-title":"Ars Combin"},{"key":"e_1_3_5_50_1","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2013.772999"},{"key":"e_1_3_5_51_1","first-page":"175","article-title":"On Minimal Edge Version of Doubly Resolving Sets of a Graph","volume":"119","author":"Ahmad M.","year":"2024","unstructured":"Ahmad, M.; Zahid, Z.; Zafar, S. On Minimal Edge Version of Doubly Resolving Sets of a Graph. JCMCC 2024, 119, 175\u2013184.","journal-title":"JCMCC"},{"key":"e_1_3_5_52_1","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2024.3357147"},{"key":"e_1_3_5_53_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10255-016-0627-0"}],"container-title":["AKCE International Journal of Graphs and Combinatorics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.tandfonline.com\/doi\/pdf\/10.1080\/09728600.2025.2487427","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T02:38:58Z","timestamp":1766111938000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.tandfonline.com\/doi\/full\/10.1080\/09728600.2025.2487427"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,22]]},"references-count":52,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,9,2]]}},"alternative-id":["10.1080\/09728600.2025.2487427"],"URL":"https:\/\/doi.org\/10.1080\/09728600.2025.2487427","relation":{},"ISSN":["0972-8600","2543-3474"],"issn-type":[{"type":"print","value":"0972-8600"},{"type":"electronic","value":"2543-3474"}],"subject":[],"published":{"date-parts":[[2025,4,22]]},"assertion":[{"value":"The publishing and review policy for this title is described in its Aims & Scope.","order":1,"name":"peerreview_statement","label":"Peer Review Statement"},{"value":"http:\/\/www.tandfonline.com\/action\/journalInformation?show=aimsScope&journalCode=uakc20","URL":"http:\/\/www.tandfonline.com\/action\/journalInformation?show=aimsScope&journalCode=uakc20","order":2,"name":"aims_and_scope_url","label":"Aim & Scope"},{"value":"2025-02-04","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-03-24","order":1,"name":"revised","label":"Revised","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-03-25","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-04-22","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}