{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T09:58:42Z","timestamp":1772013522366,"version":"3.50.1"},"reference-count":22,"publisher":"EDP Sciences","issue":"1","license":[{"start":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T00:00:00Z","timestamp":1771977600000},"content-version":"vor","delay-in-days":55,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"the National Science Foundation of China","award":["Nos.12261074 and 12461065"],"award-info":[{"award-number":["Nos.12261074 and 12461065"]}]},{"name":"Middle-aged and Young Research Fund of the Qinghai Normal University","award":["No. 2025QZR11"],"award-info":[{"award-number":["No. 2025QZR11"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2025,12,22]]},"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:p>\n                    For a family\n                    <jats:italic>\ud835\udc9c<\/jats:italic>\n                    of connected graphs, an\n                    <jats:italic>\ud835\udc9c<\/jats:italic>\n                    -factor is a spanning subgraph of a graph, whose components are isomorphic to a member of\n                    <jats:italic>\ud835\udc9c<\/jats:italic>\n                    . A graph\n                    <jats:italic>G<\/jats:italic>\n                    is called an\n                    <jats:italic>\ud835\udc9c<\/jats:italic>\n                    -factor deleted graph if\n                    <jats:italic>G<\/jats:italic>\n                    \u2013\n                    <jats:italic>e<\/jats:italic>\n                    has an\n                    <jats:italic>\ud835\udc9c<\/jats:italic>\n                    -factor for any\n                    <jats:italic>e<\/jats:italic>\n                    \u2208\n                    <jats:italic>E(G)<\/jats:italic>\n                    . A graph\n                    <jats:italic>G<\/jats:italic>\n                    is called an (\n                    <jats:italic>\ud835\udc9c<\/jats:italic>\n                    ,n)-critical deleted graph if for any V\u02ca \u2286\n                    <jats:italic>V(G)<\/jats:italic>\n                    with |V\u02ca| =\n                    <jats:italic>n<\/jats:italic>\n                    ,\n                    <jats:italic>G<\/jats:italic>\n                    \u2013\n                    <jats:italic>V<\/jats:italic>\n                    \u02ca is an\n                    <jats:italic>\ud835\udc9c<\/jats:italic>\n                    -factor deleted graph. Clearly, an (\n                    <jats:italic>\ud835\udc9c<\/jats:italic>\n                    ,0)-critical deleted graph is an\n                    <jats:italic>\ud835\udc9c<\/jats:italic>\n                    -factor deleted graph. In this paper, we write\n                    <jats:italic>\ud835\udc9c<\/jats:italic>\n                    = {\n                    <jats:italic>K<\/jats:italic>\n                    <jats:sub>2<\/jats:sub>\n                    ,\n                    <jats:italic>C<\/jats:italic>\n                    <jats:sub>\n                      2\n                      <jats:italic>i<\/jats:italic>\n                      +1\n                    <\/jats:sub>\n                    :\n                    <jats:italic>i<\/jats:italic>\n                    \u2265 2} and give several sufficient conditions for a graph to be an (\n                    <jats:italic>\ud835\udc9c<\/jats:italic>\n                    ,\n                    <jats:italic>n<\/jats:italic>\n                    )-critical deleted graph.\n                  <\/jats:p>","DOI":"10.1051\/ro\/2025169","type":"journal-article","created":{"date-parts":[[2025,12,23]],"date-time":"2025-12-23T19:50:16Z","timestamp":1766519416000},"page":"259-268","source":"Crossref","is-referenced-by-count":0,"title":["Tight toughness, isolated toughness and binding number bounds for (\n                    <i>\ud835\udc9c, n<\/i>\n                    )-critical deleted graphs"],"prefix":"10.1051","volume":"60","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-7085-9824","authenticated-orcid":false,"given":"Run","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Qinghai Normal University","place":["P.R. China"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shumin","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Qinghai Normal University","place":["P.R. China"]},{"name":"Academy of Plateau Science and Sustainability, People\u2013s Government of Qinghai Province and Beijing Normal University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chengfu","family":"Ye","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Qinghai Normal University","place":["P.R. China"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2026,2,25]]},"reference":[{"key":"R1","first-page":"97","volume":"16","author":"Akiyama","year":"1980","journal-title":"TRU Math."},{"key":"R2","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1080\/09728600.2022.2094299","volume":"19","author":"Chen","year":"2022","journal-title":"AKCE Int. J. 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