Two Problems on Interval Counting

Q3 Computer Science

Electronic Notes in Theoretical Computer Science Pub Date : 2019-08-30 DOI:10.1016/j.entcs.2019.08.055

Lívia Salgado Medeiros, Fabiano de Souza Oliveira , Jayme Luiz Szwarcfiter

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引用次数: 1

Abstract

Let $F$ be a family of intervals on the real line. An interval graph is the intersection graph of $F$ . An interval order is a partial order $(F, ≺)$ such that for all $I_{1}, I_{2} \in F$ , I₁ ≺ I₂ if and only if I₁ lies entirely at the left of I₂. Such a family $F$ is called a model of the graph (order). The interval count of a given graph (resp. order) is the smallest number of interval lengths needed in any model of this graph (resp. order). The first problem we consider is related to the classes of graphs and orders which can be represented with two interval lengths, regarding to the inclusion hierarchy among such classes. The second problem is an extremal problem which consists of determining the smallest graph or order which has interval count at least k. In particular, we study a conjecture by Fishburn on this extremal problem, verifying its validity when such a conjecture is constrained to the classes of trivially perfect orders and split orders.

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关于区间计数的两个问题

设F是实直线上的一系列区间。区间图是F的相交图。区间阶是偏阶(F， )，使得对于所有I1,I2∈F, I1 I2当且仅当I1完全位于I2的左侧。这样的族F称为图(阶)的一个模型。给定图的间隔计数。(Order)是这个图的任何模型中所需的最小的区间长度。顺序)。我们考虑的第一个问题是关于可以用两个区间长度表示的图和阶的类，以及这些类之间的包含层次。第二个问题是一个极值问题，该极值问题包括确定区间计数至少为k的最小图或阶。特别地，我们研究了Fishburn关于这个极值问题的一个猜想，并验证了该猜想在平凡完美阶和分裂阶这类条件下的有效性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Electronic Notes in Theoretical Computer Science Computer Science-Computer Science (all)

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