Abstract
We show that the class of trapezoid orders in which no trapezoid strictly contains any other trapezoid strictly contains the class of trapezoid orders in which every trapezoid can be drawn with unit area. This is different from the case of interval orders, where the class of proper interval orders is exactly the same as the class of unit interval orders.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benzer, S. (1959) On the topology of the genetic fine structure, Proc. Natl. Acad. Sci. 45, 1607–1620.
Bogart, K. P. and Isaak, G. (1996) Proper and unit bitolerance orders and graphs, Technical Report PMA-TR96–187, Dartmouth College, Hanover, NH 03755, USA.
Dagan, I., Golumbic, M. C., and Pinter, R. (1988) Trapezoid graphs and their coloring, Discrete Math. Appl. 21, 35–46.
Felsner, S. and Möhring, R. H. (1994) Semi-order dimension two is a comparability invariant, Report No. 387/1994, Technische Universität, Berlin.
Fishburn, P. and Monjardet, B. (1992) Norbert Wiener on the theory of measurement, J. Math. Psychol. 36(2), 165–184.
Fishburn, P. C. (1970) Intransitive indifference with unequal indifference intervals, J. Math. Psychol. 7, 144–149.
Fishburn, P. C. (1985) Interval Orders and Interval Graphs. Wiley, New York.
Gallai, T. (1967) Transitiv orientierbare Graphen, Acta Math. Acad. Sci. Hungar. 18, 25–66.
Gilmore, P. and Hoffman, A. (1962) Characterizations of comparability and interval graphs (abstract), in International Congress of Mathematicians, Stockholm, p. 29.
Gilmore, P. and Hoffman, A. (1964) A characterization of comparability graphs and interval graphs, Can. J. Math. 16, 539–548.
Golumbic, M. C. (1980) Algorithmic Graph Theory and Perfect graphs, Academic Press, New York.
Habib, M., Kelly, D., and Möhring, R. H. (1991) Interval dimension is a comparability invariant, Discrete Math. 88, 211–229.
Hajös, G. (1957) Ñber eine Art von Graphen, Int. Math. Nachr. 11, 65.
Langley, L. J. (1993) Interval tolerance orders and dimension: A thesis, PhD Thesis, Dartmouth College, Technical Report PMA-TR93–108, Hanover, NH 03755, USA.
Mirkin, B. G. (1972) Description of some relations on the set of real-line intervals, J. Math. Psychol. 9, 243–252.
Roberts, F. S. (1969) Indifference graphs, in F. Harary (ed.), Proof Techniques in Graph Theory, Academic Press, pp. 139–146.
Shull, R. and Trenk, A. N. (1995) Unit and proper bitolerance digraphs, Technical Report CSDTR12–1995, Wellesley College.
Wiener, N. (1914) A contribution to the theory of relative position, Proc. Camb. Philos. Soc. 17, 441–449.