Related papers: Queue Layouts of Two-Dimensional Posets
Heath and Pemmaraju conjectured that the queue-number of a poset is bounded by its width and if the poset is planar then also by its height. We show that there are planar posets whose queue-number is larger than their height, refuting the…
We investigate the queue number of posets in terms of their width, that is, the maximum number of pairwise incomparable elements. A long-standing conjecture of Heath and Pemmaraju asserts that every poset of width w has queue number at most…
The queue-number of a poset is the queue-number of its cover graph viewed as a directed acyclic graph, i.e., when the vertex order must be a linear extension of the poset. Heath and Pemmaraju conjectured that every poset of width $w$ has…
Joret et al. proved that posets with cover graphs of treewidth at most 2 have dimension at most 1276. Their proof is long and very complex. We give a short and much simpler proof that the dimension of such posets is at most 12.
In 1977, Trotter and Moore proved that a poset has dimension at most $3$ whenever its cover graph is a forest, or equivalently, has treewidth at most $1$. On the other hand, a well-known construction of Kelly shows that there are posets of…
It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such…
The classical 1991 result by Brightwell and Winkler states that the number of linear extensions of a poset is #P-complete. We extend this result to posets with certain restrictions. First, we prove that the number of linear extension for…
A linear layout of a graph $ G $ consists of a linear order $\prec$ of the vertices and a partition of the edges. A part is called a queue (stack) if no two edges nest (cross), that is, two edges $ (v,w) $ and $ (x,y) $ with $ v \prec x…
An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemer\'edi proved that any on-line algorithm could be forced to use…
Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant $d$ such that if $P$ is a poset with cover graph of $P$ of pathwidth at most $2$, then $\dim(P)\leq d$. We answer this question in the affirmative by…
A queue layout of a graph consists of a linear order on the vertices and an assignment of the edges to queues, such that no two edges in a single queue are nested. The minimum number of queues needed in a queue layout of a graph is called…
We show that height $h$ posets that have planar cover graphs have dimension $\mathcal{O}(h^6)$. Previously, the best upper bound was $2^{\mathcal{O}(h^3)}$. Planarity plays a key role in our arguments, since there are posets such that (1)…
A k-queue layout is a special type of a linear layout, in which the linear order avoids (k+1)-rainbows, i.e., k+1 independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, i.e.,…
Twin-width is a new parameter informally measuring how diverse are the neighbourhoods of the graph vertices, and it extends also to other binary relational structures, e.g. to digraphs and posets. It was introduced just very recently, in…
The dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if $P$ has dimension $d$, then to know whether $x \leq y$ in $P$ it is enough to check whether $x\leq y$ in each of the…
The dimension of a partially ordered set $P$ (poset for short) is the least positive integer $d$ such that $P$ is isomorphic to a subposet of $\mathbb{R}^d$ with the natural product order. Dimension is arguably the most widely studied…
Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every $d\geq 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$…
Let $P$ be a finite poset of width two, i.e., with no three-element antichain. We associate with $P$ a skew Young diagram $\Upsilon(P)$ and discuss some of the properties of the map $\Upsilon$. In particular, if we regard $\Upsilon(P)$ as a…
Over the past 10 years, there has been considerable interest in exploring questions connecting dimension for posets with graph theoretic properties of their cover graphs and order diagrams, especially with the concepts of planarity and…
A \emph{queue layout} of a graph consists of a \emph{linear order} of its vertices and a partition of its edges into \emph{queues}, so that no two independent edges of the same queue are nested. The \emph{queue number} of a graph is the…