7,882 research outputs found

    Contiguity orders

    Get PDF
    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

    Full text link
    The boxicity of a graph G=(V,E)G=(V,E) is the smallest integer kk for which there exist kk interval graphs Gi=(V,Ei)G_i=(V,E_i), 1ik1 \le i \le k, such that E=E1EkE=E_1 \cap \cdots \cap E_k. In the first part of this note, we prove that every graph on mm edges has boxicity O(mlogm)O(\sqrt{m \log m}), 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 GG, the boxicity of GG is at most the Colin de Verdi\`ere invariant of GG, denoted by μ(G)\mu(G). We observe that every graph GG has boxicity O(μ(G)4(logμ(G))2)O(\mu(G)^4(\log \mu(G))^2), while there are graphs GG with boxicity Ω(μ(G)logμ(G))\Omega(\mu(G)\sqrt{\log \mu(G)}). In the second part of this note, we focus on graphs embeddable on a surface of Euler genus gg. We prove that these graphs have boxicity O(glogg)O(\sqrt{g}\log g), while some of these graphs have boxicity Ω(glogg)\Omega(\sqrt{g \log g}). 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

    Full text link
    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
    corecore