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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.

05C62 Graph representations (geometric and intersection representations, etc.)
06A06 Partial orders, general
05A05 Permutations, words, matrices
Full Text: DOI
[1] 10.1137/0608005 · Zbl 0613.06002
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[5] 10.1002/jgt.3190090405 · Zbl 0665.05044
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