Unit Interval Orders of Open and Closed Intervals

TL;DR

This paper characterizes unit OC interval orders through forbidden posets and provides an efficient algorithm to recognize and construct such representations or identify obstructions.

Contribution

It offers a forbidden poset characterization and an efficient recognition algorithm for unit OC interval orders.

Findings

Provides a forbidden poset characterization.

Develops an efficient recognition algorithm.

Enables construction of interval representations or detection of obstructions.

Abstract

A poset $P = (X,\prec)$ is a unit OC interval order if there exists a representation that assigns an open or closed real interval $I(x)$ of unit length to each $x \in P$ so that $x \prec y$ in $P$ precisely when each point of $I(x)$ is less than each point in $I(y)$. In this paper we give a forbidden poset characterization of the class of unit OC interval orders and an efficient algorithm for recognizing the class. The algorithm takes a poset $P $ as input and either produces a representation or returns a forbidden poset induced in $P$.