{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T05:40:30Z","timestamp":1760334030155,"version":"build-2065373602"},"reference-count":33,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T00:00:00Z","timestamp":1753142400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>ABSTRACT<\/jats:title><jats:p>A <jats:italic>dominating<\/jats:italic> \u2010<jats:italic>model<\/jats:italic> in a graph  is a sequence  of pairwise disjoint non\u2010empty connected subgraphs of , such that for  every vertex in  has a neighbour in . Replacing \u2018every vertex in \u2019 by \u2018some vertex in \u2019 retrieves the standard definition of \u2010model, which is equivalent to  being a minor of . We explore in what sense dominating \u2010models behave like (non\u2010dominating) \u2010models. The two notions are equivalent for  but are already very different for , since the 1\u2010subdivision of any graph has no dominating \u2010model. Nevertheless, we show that every graph with no dominating \u2010model is 2\u2010degenerate and 3\u2010colourable. More generally, we prove that every graph with no dominating \u2010model is \u2010colourable. Motivated by the connection to chromatic number, we study the maximum average degree of graphs with no dominating \u2010model. We give an upper bound of  and show that random graphs provide a lower bound of , which we conjecture is asymptotically tight. This result is in contrast to the \u2010minor\u2010free setting, where the maximum average degree is . A natural strengthening of Hadwiger's conjecture arises: Is every graph with no dominating \u2010model \u2010colourable? We provide two pieces of evidence for this: (1) It is true for almost every graph. (2) Every graph  with no dominating \u2010model has a \u2010colourable induced subgraph on at least half the vertices, which implies there is an independent set of size at least .<\/jats:p>","DOI":"10.1002\/jgt.23272","type":"journal-article","created":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T07:04:14Z","timestamp":1753167854000},"page":"448-456","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dominating Kt\u2010Models"],"prefix":"10.1002","volume":"110","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5350-2379","authenticated-orcid":false,"given":"Freddie","family":"Illingworth","sequence":"first","affiliation":[{"name":"Department of Mathematics University College London London UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8866-3041","authenticated-orcid":false,"given":"David R.","family":"Wood","sequence":"additional","affiliation":[{"name":"School of Mathematics Monash University Melbourne Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,7,22]]},"reference":[{"key":"e_1_2_12_2_1","first-page":"133","article-title":"\u00dcber eine Klassifikation der Streckenkomplexe","volume":"88","author":"Hadwiger H.","year":"1943","journal-title":"Vierteljahrsschrift der Naturforschenden Gesellschaft in Z\u00fcrich"},{"key":"e_1_2_12_3_1","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-27.1.85"},{"key":"e_1_2_12_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-32162-2_13"},{"key":"e_1_2_12_5_1","unstructured":"M.DelcourtandL.Postle \u201cReducing linear Hadwiger's conjecture to coloring small graphs \u201darXiv:2108.01633 2021."},{"key":"e_1_2_12_6_1","first-page":"71","article-title":"On Hadwiger's Number and the Stability Number","volume":"13","author":"Duchet P.","year":"1982","journal-title":"Annals of Discrete Mathematics"},{"key":"e_1_2_12_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(80)80001-1"},{"key":"e_1_2_12_8_1","first-page":"37","article-title":"The Minimum Hadwiger Number for Graphs With a Given Mean Degree of Vertices","volume":"38","author":"Kostochka A. 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