Abstract
This article proves that all complete preference structures where the strict preference relation (P) has no circuit admit a representation by intervals of the real line; the rule for deciding whether an interval is indifferent or preferred to another is less straightforward than for interval orders: strict preference is indeed compatible with a certain degree of overlapping of intervals, the allowed degree being specified by means of a so-called tolerance function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abbas, M. and Vincke, Ph.: 1993, ‘Preference Structures and Threshold Models’,Journal of Multi-Criteria Decision Analysis,2, 171–178.
Anderson, C. A., Jones, K. F., Lundgren, J. R., and McKee, T. A.: 1990, ‘Competition Multigraphs and the Multicompetition Number’,Ars Combinatoria,29B, 185–192.
Berge, C.: 1973,Graphs and Hypergraphs, North-Holland, Amsterdam.
Bowser, S. and Cable, C. A.: 1991, ‘Some Recent Results on Niche Graphs’,Disc. Appl. Math.,30, 101–108.
Cable, C., Jones, K. F., Lundgren, J. R., and Seager, S.: 1989, ‘Niche Graphs’,Disc. Appl. Math.,23, 231–241.
Fishburn, P. C.: 1970, ‘Intransitive Indifference With Unequal Indifference Intervals’,J. Math. Psychol. 7, 144–149.
Fishburn, P. C.: 1985,Inteval Orders and Interval Graphs, Wiley, New York.
Fishburn, P. C., and Gehrlein, W. V.: 1992, ‘Niche Numbers’,J. Graph Theory,16, 131–139.
Golumbic, M. C. and Monma, C. L.: 1982, ‘A generalization of Interval Graphs With Tolerances’, in Proceedings 13th South Eastern Conference on Combinatorics, Graph Theory and Computing Congressus Numerantium,Utilitas Math.,35, 321–331.
Golumbic, M. C., Monma, C. L. and Trotter, W. T. Jr.: 1984, ‘Tolerance Graphs’,Disc. Appl. Math.,9, 157–170.
Isaak, G., Kim, S. R. McKee, T. A., McMorris, F. R. and Roberts, F.: 1992, ‘2-Competition Graphs’,SIAM J. Disc. Math. 5(4), 524–538.
Jacobson, M. S., McMorris, F. R. and Mulder, M.: 1991a, ‘An Introduction to Tolerance Intersection Graphs’,Proc. Sixth Int. Conf. on Theory and Applications of Graphs, Y. Alavi, G. Chartrand, O. Oellermann and A. Scwenk (Eds.) pp. 705–724.
Jacobson, M. S., McMorris, F. R. and Scheinerman, E.: 1991b, ‘General Result on Tolerance Intersection Graphs’,J. Graph Theory,15(6), 573–577.
Jones, K. F., Lundgren, J. R., Roberts, F. S., and Seager, S.: 1987 ‘Some Remarks on the Double Competition Number of a Graph’,Congressus Numerantium,60, 17–24.
Kim, S. R., McKee, T. A., McMorris, F. R. and Roberts, F. S.: ‘p-Competition Graphs’, to appear inSIAM J. Disc. Math.
Narasimhan, G. and Manber, R.: 1992, ‘Stability Number and Chromatic Number of Tolerance Graphs’,Disc. Appl. Math.,36, 47–56.
Pirlot, M.: 1990, ‘Minimal Representation of a Semiorder’,Theory and Decision,28, 109–141.
Pirlot, M.: 1991, ‘Synthetic Description of a Semiorder’,Disc. Appl. Math.,31, 299–308.
Pirlot, M. and Vincke, Ph.: 1992, ‘Lexicographic Aggregation of Semiorders’,Journal of Multi-Criteria Decision Analysis,1, 47–58.
Raychaudhuri, A., and Roberts, F. S.: 1985, ‘Generalized Competition Graphs and their Applications’, in P. Brucker and R. Pauly (Eds.),Methods of Operations Research, Vol. 49, Anton Hain, Konigstein, West Germany, pp. 295–311.
Roberts, F. S.: 1978, ‘Food Webs, Competition Graphs, and the Boxicity of Ecological phase space’, in Y. Alavi and D. Lick (Eds.),Theory and Applications of Graphs, Springer-Verlag, New York, pp. 477–490.
Roubens, M. and Vincke, Ph.: 1985,Preference Modelling. Springer-Verlag.
Roy, B.: 1980, Préférence, Indifférence, Incomparabilité,Document 9,Lamsade. Université Dauphine. Paris.
Roy, B.: 1985,Méthodologie Multicritère d'Aide à la Décision, Economica, Paris.
Roy, B. and Bouyssou, D.: 1993,Aide Multicritère à la Décision: Méthodes et cas, Economica, Paris.
Scott, D.: 1987, ‘The Competition-Common Enemy Graph of a Digraph’,Disc. Appl. Math.,17, 269–280.
Sugihara, G.: 1983, ‘Graph Theory, Homology, and Food Webs’, in S. A. Levin (Ed.),Population Biology, Proc. Sym. in Appl. Math., Vol. 30, American Mathematical Society, Providence, RI, pp. 83–101.
Vincke, Ph.: 1992,Multicriteria Decision-Aid., John Wiley & Sons, Chichester.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Abbas, M. Any complete preference structure without circuit admits an interval representation. Theor Decis 39, 115–126 (1995). https://doi.org/10.1007/BF01078980
Issue Date:
DOI: https://doi.org/10.1007/BF01078980