1,360 research outputs found
Minors and dimension
It has been known for 30 years that posets with bounded height and with cover
graphs of bounded maximum degree have bounded dimension. Recently, Streib and
Trotter proved that dimension is bounded for posets with bounded height and
planar cover graphs, and Joret et al. proved that dimension is bounded for
posets with bounded height and with cover graphs of bounded tree-width. In this
paper, it is proved that posets of bounded height whose cover graphs exclude a
fixed topological minor have bounded dimension. This generalizes all the
aforementioned results and verifies a conjecture of Joret et al. The proof
relies on the Robertson-Seymour and Grohe-Marx graph structure theorems.Comment: Updated reference
Topological minors of cover graphs and dimension
We show that posets of bounded height whose cover graphs exclude a fixed
graph as a topological minor have bounded dimension. This result was already
proven by Walczak. However, our argument is entirely combinatorial and does not
rely on structural decomposition theorems. Given a poset with large dimension
but bounded height, we directly find a large clique subdivision in its cover
graph. Therefore, our proof is accessible to readers not familiar with
topological graph theory, and it allows us to provide explicit upper bounds on
the dimension. With the introduced tools we show a second result that is
supporting a conjectured generalization of the previous result. We prove that
-free posets whose cover graphs exclude a fixed graph as a topological
minor contain only standard examples of size bounded in terms of .Comment: revised versio
On-line coloring between two lines
We study on-line colorings of certain graphs given as intersection graphs of
objects "between two lines", i.e., there is a pair of horizontal lines such
that each object of the representation is a connected set contained in the
strip between the lines and touches both. Some of the graph classes admitting
such a representation are permutation graphs (segments), interval graphs
(axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability
graphs (simple curves). We present an on-line algorithm coloring graphs given
by convex sets between two lines that uses colors on graphs with
maximum clique size .
In contrast intersection graphs of segments attached to a single line may
force any on-line coloring algorithm to use an arbitrary number of colors even
when .
The {\em left-of} relation makes the complement of intersection graphs of
objects between two lines into a poset. As an aside we discuss the relation of
the class of posets obtained from convex sets between two lines
with some other classes of posets: all -dimensional posets and all posets of
height are in but there is a -dimensional poset of height
that does not belong to .
We also show that the on-line coloring problem for curves between two lines
is as hard as the on-line chain partition problem for arbitrary posets.Comment: grant support adde
Tree-width and dimension
Over the last 30 years, researchers have investigated connections between
dimension for posets and planarity for graphs. Here we extend this line of
research to the structural graph theory parameter tree-width by proving that
the dimension of a finite poset is bounded in terms of its height and the
tree-width of its cover graph.Comment: Updates on solutions of problems and on bibliograph
Dimension of posets with planar cover graphs excluding two long incomparable chains
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 posets have two large disjoint chains with all points in
one chain incomparable with all points in the other. Gutowski and Krawczyk
conjectured that this feature is necessary. More formally, they conjectured
that for every , there is a constant such that if is a poset
with a planar cover graph and excludes , then
. We settle their conjecture in the affirmative. We also discuss
possibilities of generalizing the result by relaxing the condition that the
cover graph is planar.Comment: New section on connections with graph minors, small correction
Nowhere Dense Graph Classes and Dimension
Nowhere dense graph classes provide one of the least restrictive notions of
sparsity for graphs. Several equivalent characterizations of nowhere dense
classes have been obtained over the years, using a wide range of combinatorial
objects. In this paper we establish a new characterization of nowhere dense
classes, in terms of poset dimension: A monotone graph class is nowhere dense
if and only if for every and every , posets of height
at most with elements and whose cover graphs are in the class have
dimension .Comment: v4: Minor changes suggested by a refere
On the Duality of Semiantichains and Unichain Coverings
We study a min-max relation conjectured by Saks and West: For any two posets
and the size of a maximum semiantichain and the size of a minimum
unichain covering in the product are equal. For positive we state
conditions on and that imply the min-max relation. Based on these
conditions we identify some new families of posets where the conjecture holds
and get easy proofs for several instances where the conjecture had been
verified before. However, we also have examples showing that in general the
min-max relation is false, i.e., we disprove the Saks-West conjecture.Comment: 10 pages, 3 figure
The Queue-Number of Posets of Bounded Width or Height
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 second conjecture. On the other hand, we show that any poset of
width has queue-number at most , thus confirming the first conjecture in
the first non-trivial case. Moreover, we improve the previously best known
bounds and show that planar posets of width have queue-number at most
while any planar poset with and has queue-number at most its
width.Comment: 14 pages, 10 figures, Appears in the Proceedings of the 26th
International Symposium on Graph Drawing and Network Visualization (GD 2018
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