# Probe Interval Orders

## Document Type

Contribution to Book

## Journal/Book Title/Conference

The Mathematics of Preference,Choice and Order: Essays in Honor of Peter C. Fishburn

## Editor

S.J. Brams et al.

## Publisher

Springer-Verlag Heidelberg Berlin

## Publication Date

2009

## First Page

313

## Last Page

323

## Abstract

A probe interval graph is a graph with vertex partition *P* ∪ *N* and to each vertex *v* there corresponds an interval Iv such that vertices are adjacent if and only if their corresponding intervals intersect and at least one of the vertices belongs to *P*. If a graph has a transitive orientation on its complement, it is a cocomparability graph, and we can think of it as the incomparability graph of the order given by a transitive orientation of its complement. When the vertices of *N* have a proper representation (no interval contains another properly), a natural transitive orientation of the complement occurs. We call the order that arises a probe interval order. We characterize which probe interval graphs yield a probe interval order by restrictions placed on {Iv : v ∈ *N*}, and by the nature of the partition restricted to 4-cycles in the graph. We discuss methods for recognizing cocomparability probe interval graphs, both in the partitioned and non-partitioned case.

## Recommended Citation

Brown, D. E., L. J. Langley, "Probe Interval Orders", S.J. Brams et al. (eds.)The Mathematics of Preference, Choice and Order: Essays in Honor of Peter C. Fishburn, Springer-Verlag Heidelberg Berlin 2009.