{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T15:37:43Z","timestamp":1778686663131,"version":"3.51.4"},"reference-count":8,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":4242,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1995,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A graph <jats:italic>G<\/jats:italic> is <jats:italic>vertex domination\u2010critical<\/jats:italic> if for any vertex <jats:italic>v<\/jats:italic> of <jats:italic>G<\/jats:italic> the domination number of <jats:italic>G<\/jats:italic> \u2010 <jats:italic>v<\/jats:italic> is less than the domination number of <jats:italic>G<\/jats:italic>. If such a graph <jats:italic>G<\/jats:italic> has domination number \u03b3, it is called \u03b3\u2010critical. Brigham et al. studied \u03b3\u2010critical graphs and posed the following questions: (1) If <jats:italic>G<\/jats:italic> is a \u03b3\u2010critical graph, is |<jats:italic>V<\/jats:italic>| \u2265 (\u03b4 + 1)(\u03b3 \u2010 1) + 1?(2) If a \u03b3\u2010critical graph <jats:italic>G<\/jats:italic> has (\u0394 + 1)(\u03b3 \u2010 1) + 1 vertices, is <jats:italic>G<\/jats:italic> regular? (3) Does <jats:italic>i = \u03b3<\/jats:italic> for all \u03b3\u2010critical graphs? (4) Let <jats:italic>d<\/jats:italic> be the diameter of the \u03b3\u2010critical graph <jats:italic>G<\/jats:italic>. Does <jats:italic>d<\/jats:italic> \u2264 2(\u03b3 \u2010 1) always hold? We show that the first and third questions have a negative answer and the others have a positive answer.<\/jats:p>","DOI":"10.1002\/net.3230250203","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T19:21:57Z","timestamp":1178997717000},"page":"41-43","source":"Crossref","is-referenced-by-count":26,"title":["Vertex domination\u2010critical graphs"],"prefix":"10.1002","volume":"25","author":[{"given":"Jason","family":"Fulman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Denis","family":"Hanson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gary","family":"Macgillivray","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230180304"},{"key":"e_1_2_1_3_2","unstructured":"R. C.Brigham P. Z.Chinn andR. D.Dutton A study of vertex domination\u2010critical graphs. Department of Mathematics Technical Report M\u20102 University of Central Florida (1984)."},{"key":"e_1_2_1_4_2","unstructured":"R. C.BrighamandR. D.Dutton Diameter in vertex domination\u2010critical graphs. Manuscript University of Central Florida (1994)."},{"key":"e_1_2_1_5_2","article-title":"The diameter of domination K\u2010critical graphs","author":"Favaron O.","journal-title":"J. Graph Theory"},{"key":"e_1_2_1_6_2","volume-title":"Domination in vertex and edge critical graphs","author":"Fulman J.","year":"1992"},{"key":"e_1_2_1_7_2","first-page":"121","article-title":"Hamilton closures in domination critical graphs","volume":"13","author":"Hanson D.","year":"1993","journal-title":"J. Combin. Math. Combin. Comput."},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(83)90007-2"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190140209"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230250203","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230250203","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T01:16:20Z","timestamp":1698369380000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230250203"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,3]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1995,3]]}},"alternative-id":["10.1002\/net.3230250203"],"URL":"https:\/\/doi.org\/10.1002\/net.3230250203","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,3]]}}}