{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:25:40Z","timestamp":1759335940285},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2008,9,12]],"date-time":"2008-09-12T00:00:00Z","timestamp":1221177600000},"content-version":"unspecified","delay-in-days":5125,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[1994,9]]},"abstract":"<jats:p>A graph<jats:italic>G<\/jats:italic>is<jats:italic>threshold<\/jats:italic>if there exists a \u2018weight\u2019 function<jats:italic>w<\/jats:italic>:<jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>) \u2192<jats:italic>R<\/jats:italic>such that the total weight of any stable set of<jats:italic>G<\/jats:italic>is less than the total weight of any non-stable set of<jats:italic>G<\/jats:italic>. Let<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830000122Xxs1D4AF\" \/><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>denote the set of threshold graphs with<jats:italic>n<\/jats:italic>vertices. A graph is called<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830000122Xxs1D4AF\" \/><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-universal if it contains every threshold graph with<jats:italic>n<\/jats:italic>vertices as an induced subgraph.<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830000122Xxs1D4AF\" \/><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-universal<jats:italic>threshold<\/jats:italic>graphs are of special interest, since they are precisely those<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830000122Xxs1D4AF\" \/><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-universal graphs that do not contain any non-threshold induced subgraph.<\/jats:p><jats:p>In this paper we shall study<jats:italic>minimum<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830000122Xxs1D4AF\" \/><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-universal (threshold) graphs,<jats:italic>i.e.<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830000122Xxs1D4AF\" \/><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-universal (threshold) graphs having the minimum number of vertices.<\/jats:p><jats:p>It is shown that for any<jats:italic>n<\/jats:italic>\u2265 3 there exist minimum<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830000122Xxs1D4AF\" \/><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-universal graphs, which are themselves threshold, and others which are not.<\/jats:p><jats:p>Two extremal minimum<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830000122Xxs1D4AF\" \/><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-universal graphs having respectively the minimum and the maximum number of edges are described, it is proved that they are unique, and that they are threshold graphs.<\/jats:p><jats:p>The set of all minimum<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830000122Xxs1D4AF\" \/><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-universal threshold graphs is then described constructively; it is shown that it forms a lattice isomorphic to the<jats:italic>n<\/jats:italic>\u2212 1 dimensional Boolean cube, and that the minimum and the maximum elements of this lattice are the two extremal graphs introduced above.<\/jats:p><jats:p>The proofs provide a (polynomial) recursive procedure that determines for any threshold graph<jats:italic>G<\/jats:italic>with<jats:italic>n<\/jats:italic>vertices and for any minimum<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S096354830000122Xxs1D4AF\" \/><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-universal threshold graph<jats:italic>T<\/jats:italic>, an induced subgraph<jats:italic>G<\/jats:italic>' of<jats:italic>T<\/jats:italic>isomorphic to<jats:italic>G<\/jats:italic>.<\/jats:p>","DOI":"10.1017\/s096354830000122x","type":"journal-article","created":{"date-parts":[[2010,10,19]],"date-time":"2010-10-19T08:20:20Z","timestamp":1287476420000},"page":"327-344","source":"Crossref","is-referenced-by-count":9,"title":["On Universal Threshold Graphs"],"prefix":"10.1017","volume":"3","author":[{"given":"P. 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