{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T09:06:52Z","timestamp":1777540012891,"version":"3.51.4"},"reference-count":10,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2006,10,5]],"date-time":"2006-10-05T00:00:00Z","timestamp":1160006400000},"content-version":"vor","delay-in-days":6608,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1988,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Chv\u00e1tal defined a graph <jats:italic>G<\/jats:italic> to be brittle if each induced subgraph <jats:italic>F<\/jats:italic> of <jats:italic>G<\/jats:italic> contains a vertex that is not a midpoint of any <jats:italic>P<\/jats:italic><jats:sub>4<\/jats:sub> or not an endpoint of any <jats:italic>P<\/jats:italic><jats:sub>4<\/jats:sub>. Every brittle graph is perfectly orderable. In this paper, we prove that a graph is brittle whenever it is HHD\u2010free (containing no chordless cycle with at least five vertices, no cycle on six vertices with a long chord, and no complement of the chordless path on five vertices). We also design an <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sub>4<\/jats:sub>) algorithm to recognize HHD\u2010free graphs, and also an <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sub>4<\/jats:sub>) algorithm to construct a perfect order of an HHD\u2010free graph. It follows from this result that an optimal coloring and a largest clique of an HHD\u2010free graph can be found in <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sub>4<\/jats:sub>) time.<\/jats:p>","DOI":"10.1002\/jgt.3190120310","type":"journal-article","created":{"date-parts":[[2007,6,9]],"date-time":"2007-06-09T02:58:31Z","timestamp":1181357911000},"page":"391-404","source":"Crossref","is-referenced-by-count":37,"title":["On brittle graphs"],"prefix":"10.1002","volume":"12","author":[{"given":"C. T.","family":"Ho\u00e0ng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"N.","family":"Khouzam","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,5]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"crossref","unstructured":"C. Berge and V. Chv\u00e1tal (Eds.) Topics on Perfect Graphs. North\u2010Holland Amsterdam (1984).","DOI":"10.1016\/S0304-0208(08)72922-0"},{"key":"e_1_2_1_3_2","unstructured":"C.Berge Farbung von Graphen deren samtliche bzw. ungerade Kreise starr sind (Zusammenfassung). Math.\u2010Natur. Reihe 114 Wiss. Z. Martin\u2010Luther\u2010Univ. Halle\u2010Wintenberg (1961)."},{"key":"e_1_2_1_4_2","doi-asserted-by":"crossref","unstructured":"V.Chv\u00e1tal Perfectly ordered graphs. Topics on Perfect Graphs. North\u2010Holland Amsterdam (1984).","DOI":"10.1016\/S0304-0208(08)72923-2"},{"key":"e_1_2_1_5_2","article-title":"Four classes of perfectly orderable graphs","author":"Chv\u00e1tal V.","journal-title":"J. Graph Theory"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1137\/0603021"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02992776"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(72)90032-9"},{"key":"e_1_2_1_9_2","volume-title":"Algorithmic Graph Theory and Perfect Graphs","author":"Golumbic M. C.","year":"1980"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(79)90043-X"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(72)90006-4"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190120310","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190120310","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,27]],"date-time":"2023-09-27T05:43:11Z","timestamp":1695793391000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190120310"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1988,9]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1988,9]]}},"alternative-id":["10.1002\/jgt.3190120310"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190120310","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1988,9]]}}}