{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T04:03:28Z","timestamp":1771560208158,"version":"3.50.1"},"reference-count":13,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2021,2]]},"abstract":"<jats:p> Let [Formula: see text] be an undirected (i.e., all the edges are bidirectional), simple (i.e., no loops and multiple edges are allowed), and connected (i.e., between every pair of nodes, there exists a path) graph. Let [Formula: see text] denotes the number of edges in the shortest path or geodesic distance between two vertices [Formula: see text]. The metric dimension (or the location number) of some families of plane graphs have been obtained in [M. Imran, S. A. Bokhary and A. Q. Baig, Families of rotationally-symmetric plane graphs with constant metric dimension, Southeast Asian Bull. Math. 36 (2012) 663\u2013675] and an open problem regarding these graphs was raised that: Characterize those families of plane graphs [Formula: see text] which are obtained from the graph [Formula: see text] by adding new edges in [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, by answering this problem, we characterize some families of plane graphs [Formula: see text], which possesses the radial symmetry and has a constant metric dimension. We also prove that some families of plane graphs which are obtained from the plane graphs, [Formula: see text] by the addition of new edges in [Formula: see text] have the same metric dimension and vertices set as [Formula: see text], and only 3 nodes appropriately selected are sufficient to resolve all the nodes of these families of plane graphs. <\/jats:p>","DOI":"10.1142\/s1793830920500950","type":"journal-article","created":{"date-parts":[[2020,7,18]],"date-time":"2020-07-18T01:57:33Z","timestamp":1595037453000},"page":"2050095","source":"Crossref","is-referenced-by-count":43,"title":["Metric dimension of heptagonal circular ladder"],"prefix":"10.1142","volume":"13","author":[{"given":"Sunny Kumar","family":"Sharma","sequence":"first","affiliation":[{"name":"School of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, Jammu and Kashmir, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8423-9067","authenticated-orcid":false,"given":"Vijay Kumar","family":"Bhat","sequence":"additional","affiliation":[{"name":"School of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, Jammu and Kashmir, India"}]}],"member":"219","published-online":{"date-parts":[[2020,8,27]]},"reference":[{"key":"S1793830920500950BIB001","doi-asserted-by":"crossref","first-page":"2168","DOI":"10.1109\/JSAC.2006.884015","volume":"24","author":"Beerliova Z.","year":"2006","journal-title":"IEEE J. Selected Areas Commun."},{"key":"S1793830920500950BIB002","volume-title":"Theory and Applications of Distance Geometry","author":"Blumenthal L. M.","year":"1953"},{"issue":"1","key":"S1793830920500950BIB003","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1023\/A:1025745406160","volume":"46","author":"Buczkowski P. S.","year":"2003","journal-title":"Period. Math. Hung."},{"key":"S1793830920500950BIB004","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.endm.2005.06.023","volume":"22","author":"Caceres J.","year":"2005","journal-title":"Electron. Notes Discrete Math."},{"key":"S1793830920500950BIB005","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/S0166-218X(00)00198-0","volume":"105","author":"Chartrand G.","year":"2000","journal-title":"Discrete Appl. Math."},{"key":"S1793830920500950BIB006","first-page":"191","volume":"2","author":"Harary F.","year":"1976","journal-title":"Ars Combin."},{"key":"S1793830920500950BIB007","first-page":"663","volume":"36","author":"Imran M.","year":"2012","journal-title":"Southeast Asian Bull. Math."},{"key":"S1793830920500950BIB008","first-page":"21","volume":"75","author":"Javaid I.","year":"2008","journal-title":"Util. Math."},{"key":"S1793830920500950BIB010","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/0166-218X(95)00106-2","volume":"70","author":"Khuller S.","year":"1996","journal-title":"Discrete Appl. Math."},{"key":"S1793830920500950BIB011","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/0734-189X(84)90051-3","volume":"25","author":"Melter R. A.","year":"1984","journal-title":"Comput. Vis. Graph. Image Process."},{"key":"S1793830920500950BIB012","first-page":"549","volume-title":"Proc. 6th Southeastern Conf. Combinatorics, Graph Theory, and Computing","author":"Slater P. J.","year":"1975"},{"key":"S1793830920500950BIB013","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1007\/s00373-010-0988-8","volume":"27","author":"Tomescu I.","year":"2011","journal-title":"Graphs Combin."},{"issue":"98","key":"S1793830920500950BIB014","first-page":"371","volume":"50","author":"Tomescu I.","year":"2007","journal-title":"Bull. Math. Soc. Sci. Math. Roumanie"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830920500950","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T11:00:15Z","timestamp":1611054015000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830920500950"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,27]]},"references-count":13,"journal-issue":{"issue":"01","published-print":{"date-parts":[[2021,2]]}},"alternative-id":["10.1142\/S1793830920500950"],"URL":"https:\/\/doi.org\/10.1142\/s1793830920500950","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"value":"1793-8309","type":"print"},{"value":"1793-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,27]]}}}