Short Models for Unit Interval Graphs Min Chih Lin a,1 , Francisco J. Soulignac a,2 and Jayme L. Szwarcfiter b,3 a Departamento de Computaci´ on, FCEN, Universidad de Buenos Aires, Buenos Aires, Argentina. {oscarlin,fsoulign}@dc.uba.ar b Universidade Federal do Rio de Janeiro, Inst. de Matem´ atica, NCE and COPPE, Caixa Postal 2324, 20001-970 Rio de Janeiro, RJ, Brasil. jayme@nce.ufrj.br Abstract We present one more proof of the fact that the class of proper interval graphs is precisely the class of unit interval graphs. The proof leads to a new and efficient O(n) time and space algorithm that transforms a proper interval model of the graph into a unit model, where all the extremes are integers in the range 0 to O(n 2 ), solving a problem posed by Gardi (Discrete Math., 307 (22), 2906–2908, 2007). Keywords: algorithms and data structures, graph theory, efficient representation, proper interval graphs, unit interval graphs 1 Partially supported by UBACyT Grants X456 and X143, and PICT ANPCyT Grant 1562. 2 Partially supported by UBACyT Grant X456, PICT ANPCyT Grant 1562 and by a grant of the YPF Foundation. 3 Partially supported by the Conselho Nacional de Desenvolvimento Cient´ ıfico e Tec- nol´ ogico, CNPq, and Funda¸c˜ ao de Amparo ` a Pesquisa do Estado do Rio de Janeiro, FAPERJ, Brasil.