{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T03:18:17Z","timestamp":1773112697816,"version":"3.50.1"},"reference-count":27,"publisher":"EDP Sciences","issue":"3","license":[{"start":{"date-parts":[[2023,6,21]],"date-time":"2023-06-21T00:00:00Z","timestamp":1687305600000},"content-version":"vor","delay-in-days":51,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2023,5,29]]},"published-print":{"date-parts":[[2023,5]]},"abstract":"<jats:p>A path-factor of a graph <jats:italic>G<\/jats:italic> is a spanning subgraph of <jats:italic>G<\/jats:italic> whose components are paths. A <jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>d<\/jats:italic><\/jats:sub>-factor of a graph <jats:italic>G<\/jats:italic> is a path-factor of <jats:italic>G<\/jats:italic> whose components are paths with at least <jats:italic>d<\/jats:italic> vertices, where <jats:italic>d<\/jats:italic> is an integer with <jats:italic>d<\/jats:italic> \u2265\u00a02. A graph <jats:italic>G<\/jats:italic> is <jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>d<\/jats:italic><\/jats:sub>-factor covered if for any <jats:italic>e<\/jats:italic>\u00a0\u2208\u00a0<jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>), <jats:italic>G<\/jats:italic> admits a <jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>d<\/jats:italic><\/jats:sub>-factor including <jats:italic>e<\/jats:italic>. A graph <jats:italic>G<\/jats:italic> is (<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>d<\/jats:italic><\/jats:sub>,\u00a0<jats:italic>n<\/jats:italic>)-factor critical deleted if for any <jats:italic>Q<\/jats:italic>\u00a0\u2286\u00a0<jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>) with |<jats:italic>Q<\/jats:italic>|\u00a0=\u00a0<jats:italic>n<\/jats:italic> and any <jats:italic>e<\/jats:italic>\u00a0\u2208\u00a0<jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>\u00a0\u2212\u00a0<jats:italic>Q<\/jats:italic>), <jats:italic>G\u00a0\u2013\u00a0Q<\/jats:italic> \u2212 <jats:italic>e<\/jats:italic> has a <jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>d<\/jats:italic><\/jats:sub>-factor. A graph <jats:italic>G<\/jats:italic> is (<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>d<\/jats:italic><\/jats:sub>,\u00a0<jats:italic>n<\/jats:italic>)-factor critical covered if for any <jats:italic>Q<\/jats:italic>\u00a0\u2286\u00a0<jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>) with |<jats:italic>Q<\/jats:italic>| = <jats:italic>n<\/jats:italic>, <jats:italic>G<\/jats:italic> \u2212 <jats:italic>Q<\/jats:italic> is a <jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>d<\/jats:italic><\/jats:sub>-factor covered graph. In this paper, we verify that (1) an (<jats:italic>n<\/jats:italic>\u00a0+\u00a0<jats:italic>t<\/jats:italic>\u00a0+\u00a02)-connected graph <jats:italic>G<\/jats:italic> of order <jats:italic>p<\/jats:italic> with <jats:italic>p<\/jats:italic>\u00a0\u2265\u00a04<jats:italic>t<\/jats:italic>\u00a0+\u00a0<jats:italic>n<\/jats:italic>\u00a0+\u00a07 is (<jats:italic>P<\/jats:italic><jats:sub>\u22653<\/jats:sub>,\u00a0<jats:italic>n<\/jats:italic>)-factor critical deleted if max $ \\left\\{{d}_G({v}_1),{d}_G({v}_2),\\dots,{d}_G({v}_{2t+1})\\right\\}\\ge \\frac{p+2n}{3}$ for any independent set {<jats:italic>v<\/jats:italic><jats:sub>1<\/jats:sub>,\u00a0<jats:italic>v<\/jats:italic><jats:sub>2<\/jats:sub>,\u00a0\u2026,\u00a0<jats:italic>v<\/jats:italic><jats:sub>2<jats:italic>t<\/jats:italic>+1<\/jats:sub>} of <jats:italic>G<\/jats:italic>, where <jats:italic>n<\/jats:italic> and <jats:italic>t<\/jats:italic> are two nonnegative integers with <jats:italic>t<\/jats:italic> \u2265 1; (2) an (<jats:italic>n<\/jats:italic> + <jats:italic>t<\/jats:italic> + 1)-connected graph <jats:italic>G<\/jats:italic> of order <jats:italic>p<\/jats:italic> with <jats:italic>p<\/jats:italic>\u00a0\u2265\u00a04<jats:italic>t<\/jats:italic>\u00a0+\u00a0<jats:italic>n<\/jats:italic>\u00a0+\u00a05 is (<jats:italic>P<\/jats:italic><jats:sub>\u22653<\/jats:sub>,\u00a0<jats:italic>n<\/jats:italic>)-factor critical covered if max $ \\left\\{{d}_G({v}_1),{d}_G({v}_2),\\dots,{d}_G({v}_{2t+1})\\right\\}\\ge \\frac{p+2n+2}{3}$ for any independent set {<jats:italic>v<\/jats:italic><jats:sub>1<\/jats:sub>,\u00a0<jats:italic>v<\/jats:italic><jats:sub>2<\/jats:sub>,\u00a0\u2026,\u00a0<jats:italic>v<\/jats:italic><jats:sub>2<jats:italic>t<\/jats:italic>+1<\/jats:sub>} of <jats:italic>G<\/jats:italic>, where <jats:italic>n<\/jats:italic> and <jats:italic>t<\/jats:italic> are two nonnegative integers with <jats:italic>t<\/jats:italic>\u00a0\u2265\u00a01.<\/jats:p>","DOI":"10.1051\/ro\/2023078","type":"journal-article","created":{"date-parts":[[2023,6,2]],"date-time":"2023-06-02T19:19:35Z","timestamp":1685733575000},"page":"1443-1451","source":"Crossref","is-referenced-by-count":2,"title":["Degree conditions for path-factor critical deleted or covered graphs"],"prefix":"10.1051","volume":"57","author":[{"given":"Hongxia","family":"Liu","sequence":"first","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2023,6,21]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/S0012-365X(01)00214-X","volume":"243","author":"Ando","year":"2002","journal-title":"Discrete Math."},{"key":"R2","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/S0304-3975(00)00247-4","volume":"263","author":"Bazgan","year":"2001","journal-title":"Theor. 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