Tolerance Orders of Open and Closed Intervals

TL;DR

This paper introduces a new framework for representing posets using unit intervals with open or closed endpoints and centers, characterizing classes of such orders and providing polynomial-time recognition algorithms.

Contribution

It defines S-orders based on interval endpoint openness, characterizes several classes, and develops polynomial-time recognition algorithms for these classes.

Findings

Characterization of various S-order classes.

Examples separating different classes.

Polynomial-time recognition algorithms for S-orders.

Abstract

In this paper we combine ideas from tolerance orders with recent work on OC interval orders. We consider representations of posets by unit intervals $I_v$ in which the interval endpoints ($L(v)$ and $R(v)$) may be open or closed as well as the center point ($c(v)$). This yields four types of intervals: $A$ (endpoints and center points closed), $B$ (endpoints and center points open), $C$ (endpoints closed, center points open), and $D$ (endpoints open, center points closed). For any non-empty subset $S$ of $\{A,B,C,D\}$, we define an $S$-order as a poset $P$ that has a representation as follows: each element $v$ of $P$ is assigned a unit interval $I_v$ of type belonging to $S$, and $x \prec y$ if and only if either (i) $R(x) < c(y)$ or (ii) $R(x) = c(y)$ and at least one of $R(x), c(y)$ is open and at least one of $L(y), c(x)$ is open. We characterize several of the classes of $S$-orders and provide separating examples between unequal classes. In addition, for each $S \subseteq \{A,B,C,D\}$ we present a polynomial-time algorithm that recognizes $S$-orders, providing a representation when one exists and otherwise providing a certificate showing it is not an $S$-order.