Vol. 11, No. 5, 2018

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A simple proof characterizing interval orders with interval lengths between 1 and $k$

Simona Boyadzhiyska, Garth Isaak and Ann N. Trenk

Vol. 11 (2018), No. 5, 893–900
Abstract

A poset P = (X,) has an interval representation if each x X can be assigned a real interval Ix so that x y in P if and only if Ix lies completely to the left of Iy. Such orders are called interval orders. Fishburn (1983, 1985) 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.

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Keywords
interval order, interval graph, semiorder
Mathematical Subject Classification 2010
Primary: 05C62, 06A99
Milestones
Received: 31 August 2017
Revised: 30 January 2018
Accepted: 5 February 2018
Published: 2 April 2018

Communicated by Glenn Hurlbert
Authors
Simona Boyadzhiyska
Berlin Mathematical School
Freie Universität Berlin
Berlin
Germany
Garth Isaak
Department of Mathematics
Lehigh University
Bethlehem, PA
United States
Ann N. Trenk
Department of Mathematics
Wellesley College
Wellesley, MA
United States