{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T07:08:55Z","timestamp":1769152135585,"version":"3.49.0"},"reference-count":13,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2021,7,5]],"date-time":"2021-07-05T00:00:00Z","timestamp":1625443200000},"content-version":"vor","delay-in-days":185,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11601002"],"award-info":[{"award-number":["11601002"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2021,1]]},"abstract":"A pebbling move on a graph G<\/jats:italic> consists of the removal of two pebbles from one vertex and the placement of one pebble on an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed, which is also called the strict rubbling move. In this new move, one pebble each is removed from u<\/jats:italic> and v<\/jats:italic> adjacent to a vertex w<\/jats:italic>, and one pebble is added on w<\/jats:italic>. The rubbling number of a graph G<\/jats:italic> is the smallest number m<\/jats:italic>, such that one pebble can be moved to each vertex from every distribution with m<\/jats:italic> pebbles. The optimal rubbling number of a graph G<\/jats:italic> is the smallest number m<\/jats:italic>, such that one pebble can be moved to each vertex from some distribution with m<\/jats:italic> pebbles. In this paper, we give short proofs to determine the rubbling number of cycles and the optimal rubbling number of paths, cycles, and the grid P<\/jats:italic>2<\/jats:sub> \u00d7 P<\/jats:italic>n<\/jats:italic><\/jats:sub>; moreover, we give an upper bound of the optimal rubbling number of P<\/jats:italic>m<\/jats:italic><\/jats:sub> \u00d7 P<\/jats:italic>n<\/jats:italic><\/jats:sub>.<\/jats:p>","DOI":"10.1155\/2021\/5545080","type":"journal-article","created":{"date-parts":[[2021,7,5]],"date-time":"2021-07-05T18:20:07Z","timestamp":1625509207000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Optimal Rubbling Number of Paths, Cycles, and Grids"],"prefix":"10.1155","volume":"2021","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0173-3631","authenticated-orcid":false,"given":"Zheng-Jiang","family":"Xia","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1248-0227","authenticated-orcid":false,"given":"Zhen-Mu","family":"Hong","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2021,7,5]]},"reference":[{"key":"e_1_2_9_1_2","doi-asserted-by":"publisher","DOI":"10.1137\/0402041"},{"key":"e_1_2_9_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2012.03.010"},{"key":"e_1_2_9_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2008.09.035"},{"key":"e_1_2_9_4_2","volume-title":"Optimal t-rubbling of Complete m-ary Trees","author":"Danz L.","year":"2010"},{"key":"e_1_2_9_5_2","volume-title":"Optimal Rubbling Numbers of Graphs (In Hungarian)","author":"L\u00e1szl\u00f3 F.","year":"2010"},{"key":"e_1_2_9_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2015.10.026"},{"key":"e_1_2_9_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.endm.2011.09.079"},{"key":"e_1_2_9_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2005.03.009"},{"key":"e_1_2_9_9_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20278"},{"key":"e_1_2_9_10_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.10080"},{"key":"e_1_2_9_11_2","first-page":"65","article-title":"On pebbling graphs","volume":"107","author":"Pachter L.","year":"1995","journal-title":"The Combinatorics Numer."},{"key":"e_1_2_9_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-015-1615-5"},{"key":"e_1_2_9_13_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-020-02154-z"}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2021\/5545080.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2021\/5545080.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2021\/5545080","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,9]],"date-time":"2024-08-09T22:42:38Z","timestamp":1723243358000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2021\/5545080"}},"subtitle":[],"editor":[{"given":"M. Irfan","family":"Uddin","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":13,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1155\/2021\/5545080"],"URL":"https:\/\/doi.org\/10.1155\/2021\/5545080","archive":["Portico"],"relation":{},"ISSN":["1076-2787","1099-0526"],"issn-type":[{"value":"1076-2787","type":"print"},{"value":"1099-0526","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]},"assertion":[{"value":"2021-01-30","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-04-02","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-07-05","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"5545080"}}