{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T03:18:01Z","timestamp":1778901481498,"version":"3.51.4"},"reference-count":29,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,12]],"date-time":"2023-03-12T00:00:00Z","timestamp":1678579200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Research Projects of Department of Education of Guizhou Provincial","award":["QianJiaoHe KY[2021]250"],"award-info":[{"award-number":["QianJiaoHe KY[2021]250"]}]},{"name":"Natural Science Research Projects of Department of Education of Guizhou Provincial","award":["QianKeHe ZK[2023]279"],"award-info":[{"award-number":["QianKeHe ZK[2023]279"]}]},{"name":"Natural Science Research Projects of Department of Education of Guizhou Provincial","award":["2021JD040"],"award-info":[{"award-number":["2021JD040"]}]},{"name":"Natural Science Research Projects of Department of Education of Guizhou Provincial","award":["2020B196"],"award-info":[{"award-number":["2020B196"]}]},{"name":"Natural Science Research Projects of Department of Education of Guizhou Provincial","award":["2022BS001"],"award-info":[{"award-number":["2022BS001"]}]},{"name":"Basic Research Programs of Guizhou Province","award":["QianJiaoHe KY[2021]250"],"award-info":[{"award-number":["QianJiaoHe KY[2021]250"]}]},{"name":"Basic Research Programs of Guizhou Province","award":["QianKeHe ZK[2023]279"],"award-info":[{"award-number":["QianKeHe ZK[2023]279"]}]},{"name":"Basic Research Programs of Guizhou Province","award":["2021JD040"],"award-info":[{"award-number":["2021JD040"]}]},{"name":"Basic Research Programs of Guizhou Province","award":["2020B196"],"award-info":[{"award-number":["2020B196"]}]},{"name":"Basic Research Programs of Guizhou Province","award":["2022BS001"],"award-info":[{"award-number":["2022BS001"]}]},{"name":"Social Science Research Base Project of Department of Education of Guizhou Provincial","award":["QianJiaoHe KY[2021]250"],"award-info":[{"award-number":["QianJiaoHe KY[2021]250"]}]},{"name":"Social Science Research Base Project of Department of Education of Guizhou Provincial","award":["QianKeHe ZK[2023]279"],"award-info":[{"award-number":["QianKeHe ZK[2023]279"]}]},{"name":"Social Science Research Base Project of Department of Education of Guizhou Provincial","award":["2021JD040"],"award-info":[{"award-number":["2021JD040"]}]},{"name":"Social Science Research Base Project of Department of Education of Guizhou Provincial","award":["2020B196"],"award-info":[{"award-number":["2020B196"]}]},{"name":"Social Science Research Base Project of Department of Education of Guizhou Provincial","award":["2022BS001"],"award-info":[{"award-number":["2022BS001"]}]},{"name":"Education Science Planning Project of Guizhou Province","award":["QianJiaoHe KY[2021]250"],"award-info":[{"award-number":["QianJiaoHe KY[2021]250"]}]},{"name":"Education Science Planning Project of Guizhou Province","award":["QianKeHe ZK[2023]279"],"award-info":[{"award-number":["QianKeHe ZK[2023]279"]}]},{"name":"Education Science Planning Project of Guizhou Province","award":["2021JD040"],"award-info":[{"award-number":["2021JD040"]}]},{"name":"Education Science Planning Project of Guizhou Province","award":["2020B196"],"award-info":[{"award-number":["2020B196"]}]},{"name":"Education Science Planning Project of Guizhou Province","award":["2022BS001"],"award-info":[{"award-number":["2022BS001"]}]},{"name":"Doctoral Program of Guizhou Normal College","award":["QianJiaoHe KY[2021]250"],"award-info":[{"award-number":["QianJiaoHe KY[2021]250"]}]},{"name":"Doctoral Program of Guizhou Normal College","award":["QianKeHe ZK[2023]279"],"award-info":[{"award-number":["QianKeHe ZK[2023]279"]}]},{"name":"Doctoral Program of Guizhou Normal College","award":["2021JD040"],"award-info":[{"award-number":["2021JD040"]}]},{"name":"Doctoral Program of Guizhou Normal College","award":["2020B196"],"award-info":[{"award-number":["2020B196"]}]},{"name":"Doctoral Program of Guizhou Normal College","award":["2022BS001"],"award-info":[{"award-number":["2022BS001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The distance between two vertices of a simple connected graph G, denoted as d(u,v), is the length of the shortest path from u to v and is always symmetrical. An ordered subset W=w1,w2,w3,\u22ef,wk of V(G) is a resolving set for G, if for \u2200u,v\u2208V(G), there exists wi\u2208W \u220b d(u,wi)\u2260d(v,wi). A resolving set with minimal cardinality is called the metric basis. The metric dimension of G is the cardinality of metric basis of G and is denoted as dim(G). For the graph G1=(V1,E1,) and G2=(V2,E2), their join is denoted by G1+G2. The vertex set of G1+G2 is V1\u222aV2 and the edge set is E=E1\u222aE2\u222auv,u\u2208V1,v\u2208V2. In this article, we show that the metric dimension of the join of two path graphs is unbounded because of its dependence on the size of the paths. We also provide a general formula to determine this metric dimension. We also develop algorithms to obtain metric dimensions and a metric basis for the join of path graphs, with respect to its symmetries.<\/jats:p>","DOI":"10.3390\/sym15030708","type":"journal-article","created":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T06:09:56Z","timestamp":1678687796000},"page":"708","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Constant Time Calculation of the Metric Dimension of the Join of Path Graphs"],"prefix":"10.3390","volume":"15","author":[{"given":"Chuanjun","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics and Big Data, Guizhou Normal College, Guiyang 550018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ghulam","family":"Haidar","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, The University of Haripur, Haripur 22620, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0055-9077","authenticated-orcid":false,"given":"Murad Ul Islam","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, The University of Haripur, Haripur 22620, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Faisal","family":"Yousafzai","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences and Humanities, National University of Sciences and Technology, Islamabad 24090, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6425-2619","authenticated-orcid":false,"given":"Kostaq","family":"Hila","sequence":"additional","affiliation":[{"name":"Department of Mathematical Engineering, Polytechnic University of Tirana, 1001 Tirana, Albania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5131-577X","authenticated-orcid":false,"given":"Asad Ul Islam","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Economics, Ibn e Haldun University, Istanbul 34480, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,12]]},"reference":[{"key":"ref_1","first-page":"37","article-title":"Leaves of trees","volume":"14","author":"Slater","year":"1975","journal-title":"Congr. 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