×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 10.1137/0608005 · Zbl 0613.06002
[2] ; Doignon, Math. Sci. Humaines, 101, 49, (1988)
[3] 10.1016/0022-2496(70)90062-3 · Zbl 0191.31501
[4] 10.1137/0604031 · Zbl 0539.06003
[5] 10.1002/jgt.3190090405 · Zbl 0665.05044
[6] 10.1016/0022-2496(92)90035-6 · Zbl 0780.92029
[7] 10.1017/CBO9780511542985
[8] 10.1007/978-3-540-79128-7_17
[9] 10.1007/978-94-015-8883-6
[10] ; Schrijver, Combinatorial optimization : polyhedra and efficiency. Algorithms and Combinatorics, 24, (2003) · Zbl 1041.90001
[11] 10.2307/2964389 · Zbl 0084.24603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.