174 research outputs found
Local dimension is unbounded for planar posets
In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetřil and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height
Planar posets have dimension at most linear in their height
We prove that every planar poset of height has dimension at most
. This improves on previous exponential bounds and is best possible
up to a constant factor. We complement this result with a construction of
planar posets of height and dimension at least .Comment: v2: Minor change
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
Boolean Dimension, Components and Blocks
We investigate the behavior of Boolean dimension with respect to components
and blocks. To put our results in context, we note that for Dushnik-Miller
dimension, we have that if for every component of a poset
, then ; also if for every block
of a poset , then . By way of constrast, local dimension is
well behaved with respect to components, but not for blocks: if
for every component of a poset , then
; however, for every , there exists a poset
with and for every block of . In this
paper we show that Boolean dimension behaves like Dushnik-Miller dimension with
respect to both components and blocks: if for every
component of , then ; also if
for every block of , then .Comment: 12 pages. arXiv admin note: text overlap with arXiv:1712.0609
Ramsey properties of products of chains
Let denote the totally ordered set (or chain) on elements.
The product is a poset
called a grid. This paper discusses several loosely related results on the
Ramsey theory of grids. Most of the results involve some application of the
Product Ramsey Theorem
Twin-width I: tractable FO model checking
Inspired by a width invariant defined on permutations by Guillemot and Marx
[SODA '14], we introduce the notion of twin-width on graphs and on matrices.
Proper minor-closed classes, bounded rank-width graphs, map graphs, -free
unit -dimensional ball graphs, posets with antichains of bounded size, and
proper subclasses of dimension-2 posets all have bounded twin-width. On all
these classes (except map graphs without geometric embedding) we show how to
compute in polynomial time a sequence of -contractions, witness that the
twin-width is at most . We show that FO model checking, that is deciding if
a given first-order formula evaluates to true for a given binary
structure on a domain , is FPT in on classes of bounded
twin-width, provided the witness is given. More precisely, being given a
-contraction sequence for , our algorithm runs in time where is a computable but non-elementary function. We also prove that
bounded twin-width is preserved by FO interpretations and transductions
(allowing operations such as squaring or complementing a graph). This unifies
and significantly extends the knowledge on fixed-parameter tractability of FO
model checking on non-monotone classes, such as the FPT algorithm on
bounded-width posets by Gajarsk\'y et al. [FOCS '15].Comment: 49 pages, 9 figure
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