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A simple proof characterizing interval orders with interval lengths between 1 and $$k$$. (English) Zbl 1385.05057
Summary: A poset $$P= (X, \prec)$$ has an interval representation if each $$x \in X$$ can be assigned a real interval $$I_x$$ so that $$x \prec y$$ in $$P$$ if and only if $$I_x$$ lies completely to the left of $$I_y$$. Such orders are called interval orders. P. C. Fishburn [SIAM J. Algebraic Discrete Methods 4, 290–305 (1983; Zbl 0539.06003); Interval orders and interval graphs. A study of partially ordered sets. New York etc.: John Wiley & Sons. XI (1985; Zbl 0551.06001)] proved that for any positive integer $$k$$, an interval order has a representation in which all interval lengths are between 1 and $$k$$ if and only if the order does not contain $$(k+2)+1$$ as an induced poset. In this paper, we give a simple proof of this result using a digraph model.

MSC:
 05C62 Graph representations (geometric and intersection representations, etc.) 06A06 Partial orders, general 05A05 Permutations, words, matrices
Keywords:
interval order; interval graph; semiorder
Citations:
Zbl 0539.06003; Zbl 0551.06001
Full Text:
References:
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