{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T20:25:40Z","timestamp":1791923140904,"version":"4.3.4"},"reference-count":39,"publisher":"Oxford University Press (OUP)","issue":"9","license":[{"start":{"date-parts":[[2020,7,16]],"date-time":"2020-07-16T00:00:00Z","timestamp":1594857600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"name":"Czech Operational Programme Research, Development and Education project, International Mobility of Researchers at Charles University","award":["CZ.02.2.69\/0.0\/0.0\/16-027\/0008495"],"award-info":[{"award-number":["CZ.02.2.69\/0.0\/0.0\/16-027\/0008495"]}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2018M631322"],"award-info":[{"award-number":["2018M631322"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971054"],"award-info":[{"award-number":["11971054"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11731002"],"award-info":[{"award-number":["11731002"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"111 Project of China","award":["B16002"],"award-info":[{"award-number":["B16002"]}]},{"DOI":"10.13039\/100007225","name":"Ministry of Science and Technology","doi-asserted-by":"publisher","award":["107-2221-E-141-001-MY3"],"award-info":[{"award-number":["107-2221-E-141-001-MY3"]}],"id":[{"id":"10.13039\/100007225","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,9,17]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Given a connected graph $G$ and a positive integer $\\ell $, the $\\ell $-extra (resp. $\\ell $-component) edge connectivity of $G$, denoted by $\\lambda ^{(\\ell )}(G)$ (resp. $\\lambda _{\\ell }(G)$), is the minimum number of edges whose removal from $G$ results in a disconnected graph so that every component has more than $\\ell $ vertices (resp. so that it contains at least $\\ell $ components). This naturally generalizes the classical edge connectivity of graphs defined in term of the minimum edge cut. In this paper, we proposed a general approach to derive component (resp. extra) edge connectivity for a connected graph $G$. For a connected graph $G$, let $S$ be a vertex subset of $G$ for $G\\in \\{\\Gamma _{n}(\\Delta ),AG_n,S_n^2\\}$ such that $|S|=s\\leq |V(G)|\/2$, $G[S]$ is connected and $|E(S,G-S)|=\\min \\limits _{U\\subseteq V(G)}\\{|E(U, G-U)|: |U|=s, G[U]\\ \\textrm{is connected}\\ \\}$, then we prove that $\\lambda ^{(s-1)}(G)=|E(S,G-S)|$ and $\\lambda _{s+1}(G)=|E(S,G-S)|+|E(G[S])|$ for $s=3,4,5$. By exploring the reliability analysis of $AG_n$ and $S_n^2$ based on extra (component) edge faults, we obtain the following results: (i) $\\lambda _3(AG_n)-1=\\lambda ^{(1)}(AG_n)=4n-10$, $\\lambda _4(AG_n)-3=\\lambda ^{(2)}(AG_n)=6n-18$ and $\\lambda _5(AG_n)-4=\\lambda ^{(3)}(AG_n)=8n-24$; (ii) $\\lambda _3(S_n^2)-1=\\lambda ^{(1)}(S_n^2)=4n-8$, $\\lambda _4(S_n^2)-3=\\lambda ^{(2)}(S_n^2)=6n-15$ and $\\lambda _5(S_n^2)-4=\\lambda ^{(3)}(S_n^2)=8n-20$. This general approach maybe applied to many diverse networks.<\/jats:p>","DOI":"10.1093\/comjnl\/bxaa070","type":"journal-article","created":{"date-parts":[[2020,6,3]],"date-time":"2020-06-03T19:09:50Z","timestamp":1591211390000},"page":"1425-1436","source":"Crossref","is-referenced-by-count":14,"title":["Reliability Analysis of Alternating Group Graphs and Split-Stars"],"prefix":"10.1093","volume":"64","author":[{"given":"Mei-Mei","family":"Gu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China"},{"name":"Faculty of Mathematics and Physics, Charles University, Prague 11800, Czech Republic"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rong-Xia","family":"Hao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jou-Ming","family":"Chang","sequence":"additional","affiliation":[{"name":"Institute of Information and Decision Sciences, National Taipei University of Business, Taipei, 10051, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2020,7,16]]},"reference":[{"key":"2021091609314517500_ref1","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1002\/net.3230130303","article-title":"Conditional connectivity","volume":"13","author":"Harary","year":"1983","journal-title":"Networks"},{"key":"2021091609314517500_ref2","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/0012-365X(94)00369-T","article-title":"On the extra connectivity graphs","volume":"155","author":"F\u00e1brega","year":"1996","journal-title":"Discrete Math."},{"key":"2021091609314517500_ref3","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/j.ipl.2005.02.001","article-title":"Super connectivity of line graphs","volume":"94","author":"Xu","year":"2005","journal-title":"Inform. 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