Tree-width and dimension

Joret, Gwenaël; Micek, Piotr; Milans, Kevin G.; Trotter, William T.; Walczak, Bartosz; Wang, Ruidong

doi:10.1007/s00493-014-3081-8

Cited by 28 publications

(32 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For fixed genus, this is a 2 O(h 3 ) bound on the dimension. In particular, this improves on the previous best bound for posets with planar cover graphs [9], which was doubly exponential in the height.…”

Section: Applicationssupporting

confidence: 52%

“…For fixed t, this is a 2 O(h t ) bound on the dimension, which improves on the doubly exponentional bound in [9]. Surprisingly, this upper bound turns out to be essentially best possible:…”

Section: Applicationsmentioning

confidence: 76%

“…• have bounded treewidth, bounded genus, or more generally exclude an apex-graph as minor [9]; • exclude a fixed graph as a (topological) minor [28,17];…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nowhere Dense Graph Classes and Dimension

et al. 2019

Self Cite

View full text Add to dashboard Cite

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 h 1 and every ε > 0, posets of height at most h with n elements and whose cover graphs are in the class have dimension O(n ε ).

show abstract

Section: Applicationssupporting

confidence: 52%

“…For fixed t, this is a 2 O(h t ) bound on the dimension, which improves on the doubly exponentional bound in [9]. Surprisingly, this upper bound turns out to be essentially best possible:…”

Section: Applicationsmentioning

confidence: 76%

See 1 more Smart Citation

Nowhere Dense Graph Classes and Dimension

et al. 2019

Self Cite

View full text Add to dashboard Cite

show abstract

“…We encourage readers to consult the discussion of connections between Dushnik-Miller dimension and structural graph theory as detailed in [15], [36] and [35]. Here we provide a quick summary of highlights.…”

Section: Connections With Structural Graph Theorymentioning

confidence: 99%

“…The first major result linking dimension and structural graph theory is due to Joret, Micek, Milans, Trotter, Walczak and Wang [15], who proved that the dimension of a poset is bounded as a function of its height and the tree-width of its cover graph. More formally, they showed that for each pair (t, h) of positive integers, there is a least positive integer d(t, h) so that if P is a poset of height h and the tree-width of the cover graph of P is t, then dim(P ) ≤ d(t, h).…”

Section: Connections With Structural Graph Theorymentioning

confidence: 99%

Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

Barrera-Cruz

Prag

Smith

et al. 2019

Order

Self Cite

View full text Add to dashboard Cite

The original notion of dimension for posets is due to Dushnik and Miller and has been studied extensively in the literature. Quite recently, there has been considerable interest in two variations of dimension known as Boolean dimension and local dimension. For a poset P , the Boolean dimension of P and the local dimension of P are both bounded from above by the dimension of P and can be considerably less. Our primary goal will be to study analogies and contrasts among these three parameters. As one example, it is known that the dimension of a poset is bounded as a function of its height and the tree-width of its cover graph. The Boolean dimension of a poset is bounded in terms of the tree-width of its cover graph, independent of its height. We show that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of height. We also prove that the local dimension of a poset is bounded in terms of the path-width of its cover graph. In several of our results, Ramsey theoretic methods will be applied.

show abstract

Topological Minors of Cover Graphs and Dimension

Micek

Wiechert

2017

Journal of Graph Theory

Self Cite

View full text Add to dashboard Cite

Abstract. 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 (k + k)-free posets whose cover graphs exclude a fixed graph as a topological minor contain only standard examples of size bounded in terms of k.

show abstract

Tree-width and dimension

Cited by 28 publications

References 36 publications

Nowhere Dense Graph Classes and Dimension

Nowhere Dense Graph Classes and Dimension

Comparing Dushnik-Miller Dimension, Boolean Dimension and Local Dimension

Topological Minors of Cover Graphs and Dimension

Contact Info

Product

Resources

About