{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:47:04Z","timestamp":1777704424376,"version":"3.51.4"},"reference-count":19,"publisher":"SAGE Publications","issue":"4","license":[{"start":{"date-parts":[[2018,8,8]],"date-time":"2018-08-08T00:00:00Z","timestamp":1533686400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent & Fuzzy Systems"],"published-print":{"date-parts":[[2018,10,27]]},"abstract":"\n Network function virtualization (NFV) can be regarded as the latest trick development in the provisioning of network service. Software programs are running on virtual machines and industry standard servers to replace the traditional hardware middleboxes, and thus lead to flexibility, service agility and cost decreasing. A basic problem in NFV service chain provisioning is the ability of resource scheduling which equals to the existence of fractional factor. The concept of all fractional (\n g<\/jats:italic>\n ,\n f<\/jats:italic>\n ,\n n<\/jats:italic>\n \u2032,\n m<\/jats:italic>\n )-critical deleted graph is the extension of fractional (\n g<\/jats:italic>\n ,\n f<\/jats:italic>\n ,\n n<\/jats:italic>\n \u2032,\n m<\/jats:italic>\n )-critical deleted graph. In this paper, we consider the resource scheduling problem in NFV networks using graph theory, and an independent set degree condition and an independent set neighborhood union condition for all fractional (\n g<\/jats:italic>\n ,\n f<\/jats:italic>\n ,\n n<\/jats:italic>\n \u2032,\n m<\/jats:italic>\n )-critical deleted graphs are determined. Furthermore, we show that the result are tight on independent set condition.\n <\/jats:p>","DOI":"10.3233\/jifs-169768","type":"journal-article","created":{"date-parts":[[2018,8,14]],"date-time":"2018-08-14T15:54:00Z","timestamp":1534262040000},"page":"4495-4502","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Independent set conditions for all fractional (\n g<\/i>\n ,\n f<\/i>\n ,\n n<\/i>\n \u2032,\n m<\/i>\n )-critical deleted NFV networks"],"prefix":"10.1177","volume":"35","author":[{"given":"Jianzhang","family":"Wu","sequence":"first","affiliation":[{"name":"College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China"}]},{"given":"Jiabin","family":"Yuan","sequence":"additional","affiliation":[{"name":"College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China"}]},{"given":"Muhammad Kamran","family":"Siddiqui","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Comsats Institute of Information Technology, Sahiwal, Pakistan"}]}],"member":"179","published-online":{"date-parts":[[2018,8,8]]},"reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84628-970-5"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2013.02.006"},{"key":"e_1_3_2_4_2","doi-asserted-by":"crossref","unstructured":"GaoW. and GaoY. Toughness condition for a graph to be a fractional (g f n)-critical deleted graph The Scientific World Journal2014 7. Article ID 369798. 10.1155\/2014-369798.","DOI":"10.1155\/2014\/369798"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10114-014-2629-2"},{"key":"e_1_3_2_6_2","doi-asserted-by":"publisher","DOI":"10.4064\/cm6713-8-2016"},{"key":"e_1_3_2_7_2","doi-asserted-by":"publisher","DOI":"10.4064\/cm6959-8-2016"},{"key":"e_1_3_2_8_2","first-page":"295","article-title":"Toughness and fractional critical deleted graph","volume":"98","author":"Gao W.","year":"2015","unstructured":"GaoW. and WangW.F., Toughness and fractional critical deleted graph, Utilitas Mathematica98 (2015), 295\u2013310.","journal-title":"Utilitas Mathematica"},{"key":"e_1_3_2_9_2","first-page":"225","article-title":"A neighborhood union condition for fractional (k, m)-deleted graphs","volume":"113","author":"Gao W.","year":"2014","unstructured":"GaoW. and WangW.F., A neighborhood union condition for fractional (k, m)-deleted graphs, Ars Combinatoria113 (2014), 225\u2013233.","journal-title":"Ars Combinatoria"},{"key":"e_1_3_2_10_2","article-title":"Two tight independent set conditions for fractional (g, f, m)-deleted graphs systems","author":"Gao W.","unstructured":"GaoW., GuiraoJ.L.G. and WuH.L., Two tight independent set conditions for fractional (g, f, m)-deleted graphs systems, Qualitative Theory of Dynamical SystemsDOI: 10.1007\/s12346-016-0222-z.","journal-title":"Qualitative Theory of Dynamical Systems"},{"key":"e_1_3_2_11_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40840-015-0194-1"},{"key":"e_1_3_2_12_2","doi-asserted-by":"publisher","DOI":"10.4134\/JKMS.2014.51.1.055"},{"key":"e_1_3_2_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2015.11.035"},{"key":"e_1_3_2_14_2","first-page":"303","article-title":"An isolated toughness condition for graphs to be fractional (k, m)-deleted graphs","volume":"105","author":"Gao W.","year":"2017","unstructured":"GaoW., LiangL. and ChenY.H., An isolated toughness condition for graphs to be fractional (k, m)-deleted graphs, Utilitas Mathematica105 (2017), 303\u2013316.","journal-title":"Utilitas Mathematica"},{"key":"e_1_3_2_15_2","first-page":"377","article-title":"A sufficient condition for a graph to be fractional (a, b, n)-critical deleted graph","volume":"119","author":"Gao W.","year":"2015","unstructured":"GaoW., A sufficient condition for a graph to be fractional (a, b, n)-critical deleted graph, Ars Combinatoria119 (2015), 377\u2013390.","journal-title":"Ars Combinatoria"},{"key":"e_1_3_2_16_2","doi-asserted-by":"publisher","DOI":"10.21042\/AMNS.2016.2.00047"},{"key":"e_1_3_2_17_2","doi-asserted-by":"publisher","DOI":"10.21042\/AMNS.2017.2.00044"},{"key":"e_1_3_2_18_2","doi-asserted-by":"publisher","DOI":"10.21042\/AMNS.2017.2.00046"},{"key":"e_1_3_2_19_2","unstructured":"GaoW. 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Neighborhood condition for all fractional (g f n\u2019 m)-critical deleted graphs Open Physics In press."},{"key":"e_1_3_2_20_2","first-page":"53","article-title":"A note on fractional (g, f, n\u2019, m)-critical deleted graph","volume":"1","author":"Gao W.","year":"2012","unstructured":"GaoW. and YanC.C., A note on fractional (g, f, n\u2019, m)-critical deleted graph, Advances in Computational Mathematics and its Applications1 (2012), 53\u201355.","journal-title":"Advances in Computational Mathematics and its Applications"}],"container-title":["Journal of Intelligent & Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-169768","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/JIFS-169768","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-169768","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:41:11Z","timestamp":1777455671000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/JIFS-169768"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,8]]},"references-count":19,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2018,10,27]]}},"alternative-id":["10.3233\/JIFS-169768"],"URL":"https:\/\/doi.org\/10.3233\/jifs-169768","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,8,8]]}}}