Abstract
Abstract A unit interval graph (respectively, n-graph) is the intersection graph of a family of intervals of unit length (respectively, of sets of n consecutive integers). Roberts (Annals of the New York Academy of Science 319 (1979) 466–483) showed that these two classes of graphs are essentially the same, by using a notion of compatibility similar to the one that appeared long ago in Goodman (The Structure of Appearance (Harvard University Press, 1951)) and in Fine and Harop (J. Symbolic Logic 22 (1957) 130–140) when they studied the concept of linear order in finite linear arrays arising in applications to perceptual problems. We give a linear time algorithm to find the minimum n such that a given unit interval graph is an n-graph in a particular case. A linear programming formulation is also presented. We conclude with a discussion of the connections between this work and the theory of semiorders.
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