Ann N Trenk | Searcharxiv

arXiv2017

Interval Orders with Two Interval Lengths

A poset $P = (X,\prec)$ has an interval representation if each $x \in X$ can be assigned a real interval $I_x$ so that $x \prec y$ in $P$ if and only if $I_x$ lies completely to the left of $I_y$. Such orders are called \emph{interval orders}. In this paper we give a surprisingly simple forbidden poset characterization of those posets that have an interval representation in which each interval length is either 0 or 1. In addition, for posets $(X,\prec)$ with a weight of 1 or 2 assigned to each point, we characterize those that have an interval representation in which for each $x \in X$ the length of the interval assigned to $x$ equals the weight assigned to $x$. For both these problems we can determine in polynomial time whether the desired interval representation is possible and in the affirmative case, produce such a representation.

math.CO↗

arXiv2017

Tolerance Orders of Open and Closed Intervals

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.

math.CO↗