{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T04:16:59Z","timestamp":1780633019593,"version":"3.54.1"},"reference-count":35,"publisher":"EDP Sciences","issue":"5","license":[{"start":{"date-parts":[[2024,10,24]],"date-time":"2024-10-24T00:00:00Z","timestamp":1729728000000},"content-version":"vor","delay-in-days":53,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2024,9,11]]},"published-print":{"date-parts":[[2024,9]]},"abstract":"<jats:p>For any graph <jats:italic>G<\/jats:italic>, a subset <jats:italic>D \u2286 V<\/jats:italic> (<jats:italic>G<\/jats:italic>) is called a <jats:italic>P<\/jats:italic><jats:sub>3<\/jats:sub>-isolating set of <jats:italic>G<\/jats:italic> if <jats:italic>G<\/jats:italic> \u2212 <jats:italic>N<\/jats:italic>[<jats:italic>D<\/jats:italic>] contains no <jats:italic>P<\/jats:italic><jats:sub>3<\/jats:sub> as a subgraph, that is, consists of isolated vertices and isolated edges only. The <jats:italic>P<\/jats:italic><jats:sub>3<\/jats:sub>-isolation number of <jats:italic>G<\/jats:italic>, denoted by <jats:italic>\u03b9<\/jats:italic>(<jats:italic>G, P<\/jats:italic><jats:sub>3<\/jats:sub>), is the cardinality of a smallest <jats:italic>P<\/jats:italic><jats:sub>3<\/jats:sub>-isolating set of <jats:italic>G<\/jats:italic>. Zhang and Wu [<jats:italic>Discrete Appl. Math<\/jats:italic>. 304 (2021) 365\u2013374] investigated the parameter <jats:italic>\u03b9<\/jats:italic>(<jats:italic>G, P<\/jats:italic><jats:sub>3<\/jats:sub>) of a graph, and they proved that if <jats:italic>G<\/jats:italic> \u2209 {<jats:italic>P<\/jats:italic><jats:sub>3<\/jats:sub>, <jats:italic>C<\/jats:italic><jats:sub>3<\/jats:sub>, <jats:italic>C<\/jats:italic><jats:sub>6<\/jats:sub>} is a connected graph of order <jats:italic>n<\/jats:italic>, then <jats:italic>\u03b9<\/jats:italic>(<jats:italic>G,P<\/jats:italic>3)\u226427<jats:italic>n<\/jats:italic>. In this paper, we shall prove that if <jats:italic>G<\/jats:italic> \u2209 {<jats:italic>P<\/jats:italic><jats:sub>3<\/jats:sub>, <jats:italic>C<\/jats:italic><jats:sub>7<\/jats:sub>, <jats:italic>C<\/jats:italic><jats:sub>11<\/jats:sub>} is a connected graph of order <jats:italic>n<\/jats:italic> without triangles and induced 6-cycles, then <jats:italic>\u03b9<\/jats:italic>(<jats:italic>G,P<\/jats:italic>3)\u2264<jats:italic>n<\/jats:italic>4, and the upper bound is sharp. This extends a result on <jats:italic>\u03b9<\/jats:italic>(<jats:italic>T, P<\/jats:italic><jats:sub>3<\/jats:sub>) of a tree <jats:italic>T<\/jats:italic> by Caro and Hansberg [<jats:italic>Filomat<\/jats:italic> 31 (2017) 3925\u20133944].<\/jats:p>","DOI":"10.1051\/ro\/2024178","type":"journal-article","created":{"date-parts":[[2024,9,12]],"date-time":"2024-09-12T19:01:51Z","timestamp":1726167711000},"page":"4607-4619","source":"Crossref","is-referenced-by-count":2,"title":["A note on the <i>P<\/i><sub>3<\/sub>-isolation number of a graph"],"prefix":"10.1051","volume":"58","author":[{"given":"Xiaohua","family":"Wei","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0602-536X","authenticated-orcid":false,"given":"Gang","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Biao","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"250","published-online":{"date-parts":[[2024,10,24]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"3367","DOI":"10.1051\/ro\/2022150","volume":"56","author":"Abd Aziz","year":"2022","journal-title":"RAIRO:RO"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"Bollob\u00e1s B., Modern Graph Theory. Springer, New York (1998).","DOI":"10.1007\/978-1-4612-0619-4"},{"key":"R3","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1002\/jgt.3190030306","volume":"3","author":"Bollob\u00e1s","year":"1979","journal-title":"J. Graph Theory"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"Bondy J.A. and Murty U.S.R., Graph Theory. Springer, New York (2008).","DOI":"10.1007\/978-1-84628-970-5"},{"key":"R5","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1007\/s00373-020-02143-2","volume":"36","author":"Borg","year":"2020","journal-title":"Graphs Combin."},{"key":"R6","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.dam.2023.05.004","volume":"339","author":"Borg","year":"2023","journal-title":"Discrete Appl. Math."},{"key":"R7","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1016\/j.dam.2020.01.005","volume":"283","author":"Borg","year":"2020","journal-title":"Discrete Appl. Math."},{"key":"R8","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s00026-022-00620-4","volume":"27","author":"Borg","year":"2023","journal-title":"Ann. Comb."},{"key":"R9","doi-asserted-by":"crossref","first-page":"111879","DOI":"10.1016\/j.disc.2020.111879","volume":"343","author":"Borg","year":"2020","journal-title":"Discrete Math."},{"key":"R10","doi-asserted-by":"crossref","first-page":"112641","DOI":"10.1016\/j.disc.2021.112641","volume":"345","author":"Borg","year":"2022","journal-title":"Discrete Math."},{"key":"R11","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1016\/j.dam.2012.08.023","volume":"161","author":"Campos","year":"2013","journal-title":"Discrete Appl. Math."},{"key":"R12","doi-asserted-by":"crossref","first-page":"3925","DOI":"10.2298\/FIL1712925C","volume":"31","author":"Caro","year":"2017","journal-title":"Filomat"},{"key":"R13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00373-011-1040-3","volume":"28","author":"Chellali","year":"2012","journal-title":"Graphs Combin."},{"key":"R14","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.akcej.2017.11.005","volume":"16","author":"Chelvam","year":"2019","journal-title":"AKCE Int. J. Graphs Comb."},{"key":"R15","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1016\/j.dam.2023.07.018","volume":"340","author":"Chen","year":"2023","journal-title":"Discrete Appl. Math."},{"key":"R16","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1002\/jgt.22968","volume":"104","author":"Cho","year":"2023","journal-title":"J. Graph Theory"},{"key":"R17","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/s00373-023-02717-w","volume":"39","author":"Cui","year":"2023","journal-title":"Graphs Combin."},{"key":"R18","first-page":"101","volume":"94","author":"Desormeaux","year":"2014","journal-title":"Util. Math."},{"key":"R19","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1002\/jgt.21855","volume":"80","author":"Dorbec","year":"2015","journal-title":"J. Graph Theory"},{"key":"R20","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.dam.2020.09.011","volume":"289","author":"Favaron","year":"2021","journal-title":"Discrete Appl. Math."},{"key":"R21","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1016\/j.disc.2012.11.031","volume":"313","author":"Goddard","year":"2013","journal-title":"Discrete Math."},{"key":"R22","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1137\/11082574","volume":"25","author":"Gon\u00e7alves","year":"2011","journal-title":"SIAM J. Discrete Math."},{"key":"R23","unstructured":"Haynes T.W., Hedetniemi S.T. and Slater P.J., Fundamentals of Domination in Graphs. Marcel Dekker Inc., New York (1998)."},{"key":"R24","doi-asserted-by":"crossref","unstructured":"Haynes T.W., Hedetniemi S.T. and Henning M.A., editors, Structures of domination in graphs. In Vol. 66 Developments in Mathematics. Springer, Cham (2021).","DOI":"10.1007\/978-3-030-58892-2"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"Haynes T.W., Hedetniemi S.T. and Henning M.A., Domination in Graphs: Core Concepts. Springer Monographs in Mathematics. Springer, Cham (2023).","DOI":"10.1007\/978-3-031-09496-5"},{"key":"R26","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.disc.2007.12.044","volume":"309","author":"Henning","year":"2009","journal-title":"Discrete Math."},{"key":"R27","doi-asserted-by":"crossref","first-page":"112965","DOI":"10.1016\/j.disc.2022.112965","volume":"345","author":"Hua","year":"2022","journal-title":"Discrete Math."},{"key":"R28","doi-asserted-by":"crossref","first-page":"S853","DOI":"10.1051\/ro\/2020025","volume":"55","author":"Khoeilar","year":"2021","journal-title":"RAIRO:RO"},{"key":"R29","doi-asserted-by":"crossref","first-page":"2090","DOI":"10.1137\/120884742","volume":"27","author":"Kinnersley","year":"2013","journal-title":"SIAM J. Discrete Math."},{"key":"R30","doi-asserted-by":"crossref","first-page":"113086","DOI":"10.1016\/j.disc.2022.113086","volume":"345","author":"Kosari","year":"2022","journal-title":"Discrete Math."},{"key":"R31","doi-asserted-by":"crossref","unstructured":"Ore O., Theory of Graphs. In Vol. 38 American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (1962).","DOI":"10.1090\/coll\/038"},{"key":"R32","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.dam.2019.06.011","volume":"267","author":"Tokunaga","year":"2019","journal-title":"Discrete Appl. Math."},{"key":"R33","doi-asserted-by":"crossref","first-page":"1169","DOI":"10.1007\/s40840-022-01248-6","volume":"45","author":"Yan","year":"2022","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"R34","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1016\/j.dam.2021.08.013","volume":"304","author":"Zhang","year":"2021","journal-title":"Discrete Appl. Math."},{"key":"R35","first-page":"169","volume":"39","author":"Zhang","year":"2022","journal-title":"J. Xinjiang Univ. (Nat. Sci. Ed. Chin. Eng.)"}],"container-title":["RAIRO - Operations Research"],"original-title":[],"link":[{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2024178\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,24]],"date-time":"2024-10-24T08:01:21Z","timestamp":1729756881000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2024178"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9]]},"references-count":35,"journal-issue":{"issue":"5"},"alternative-id":["ro230859"],"URL":"https:\/\/doi.org\/10.1051\/ro\/2024178","relation":{},"ISSN":["0399-0559","2804-7303"],"issn-type":[{"value":"0399-0559","type":"print"},{"value":"2804-7303","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9]]}}}