7,882 research outputs found
Contiguity orders
This paper is devoted to the study of contiguity orders i.e. orders having a linear extension extension L such that all upper (or lower) cover sets are intervals of L. This new class is a strict generalization of both interval orders and N-free orders and is linearly recognizable. It is proved that computing the number of contiguity extensions is #P-complete and that the dimension of height one contiguity orders is polynomially tractable. Moreover the membership is a comparability invariant on bi-contiguity orders. Finally for strong-contiguity orders the calculation of the dimension is NP-complete
Boxicity and topological invariants
The boxicity of a graph is the smallest integer for which there
exist interval graphs , , such that . In the first part of this note, we prove that every graph on
edges has boxicity , which is asymptotically best
possible. We use this result to study the connection between the boxicity of
graphs and their Colin de Verdi\`ere invariant, which share many similarities.
Known results concerning the two parameters suggest that for any graph , the
boxicity of is at most the Colin de Verdi\`ere invariant of , denoted by
. We observe that every graph has boxicity , while there are graphs with boxicity . In the second part of this note, we focus on graphs embeddable on a
surface of Euler genus . We prove that these graphs have boxicity
, while some of these graphs have boxicity . This improves the previously best known upper and lower bounds.
These results directly imply a nearly optimal bound on the dimension of the
adjacency poset of graphs on surfaces.Comment: 6 page
Multiple testing with persistent homology
Multiple hypothesis testing requires a control procedure. Simply increasing
simulations or permutations to meet a Bonferroni-style threshold is
prohibitively expensive. In this paper we propose a null model based approach
to testing for acyclicity, coupled with a Family-Wise Error Rate (FWER) control
method that does not suffer from these computational costs. We adapt an False
Discovery Rate (FDR) control approach to the topological setting, and show it
to be compatible both with our null model approach and with previous approaches
to hypothesis testing in persistent homology. By extending a limit theorem for
persistent homology on samples from point processes, we provide theoretical
validation for our FWER and FDR control methods
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