An algorithm for estimating the crossing number of dense graphs, and continuous analogs of the crossing and rectilinear crossing numbers

We present a deterministic $n^{2+o(1)}$-time algorithm that approximates the crossing number of any graph $G$ of order $n$ up to an additive error of $o(n^4)$. We also provide a randomized polynomial-time algorithm that constructs a drawing of $G$ with $\text{cr}(G)+o(n^4)$ crossings. These results are made interesting by the well known fact that every dense $n$-vertex graph has crossing number $\Theta(n^4)$. Our work builds on a technique developed by Fox, Pach and S\'uk, who obtained very similar results for the rectilinear crossing number. The results by the aforementioned authors and in this paper imply that the (normalized) crossing and rectilinear crossing numbers are estimable parameters. Motivated by this, we introduce two graphon parameters, the crossing density and the rectilinear crossing density, and then we prove that, in a precise sense, these are the correct continuous analogs of the crossing and rectilinear crossing numbers of graphs.Comment: 23 pages, 4 figure

Cerebral gliosarcoma with perivascular involvement in a cat

Case summary A 5-year-old neutered male domestic shorthair cat presented with an 18-month history of facial tics, and progressive general ataxia, weakness, lethargy and anorexia of 2 weeks' duration. MRI of the brain showed a well-defined heterogeneous hyperintense mass on T1-weighted and T2-weighted images, with central hypointensity in the rostral commissure and septum pellucidum, and perilesional hyperintensity in fluid-attenuated inversion recovery, suggestive of perilesional oedema. Gross examination in a transverse section of the brain at the level of the septum pellucidum revealed a 0.2 cm brown soft mass. Histopathological examination identified a biphasic neoplastic proliferation of mesenchymal and neuroepithelial cell populations. Fusiform cells were predominately distributed in bundles showing a high degree of anisocytosis and marked immune-positive reaction to vimentin immunochemistry, confirming a sarcomatous origin. Additionally, high numbers of astrocytic cells were identified by an intense immunopositive reaction to glial fibrillary acidic protein and negative reaction to oligodendrocyte transcription factor 2 immunochemistry. Vascular invasion of the neoplasia into the wall of a medium branch of the rostral cerebral artery was present (secondary Scherer structures). Based on these characteristics, the tumour was defined as a gliosarcoma. Gliosarcoma is a recognised astrocytoma grade IV anaplastic glial cell tumour with sarcomatous differentiation. Relevance and novel information To our knowledge, this is the first report describing a cerebral gliosarcoma in a cat including clinical, MRI, macroscopic and histopathological features and immunolabelling characteristics

Separability, Boxicity, and Partial Orders

This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/A collection S = {Si,..., Sn} of disjoint closed convex sets in Rd is separable if there exists a direction (a non-zero vector) −→v of Rd such that the elements of S can be removed, one at a time, by translating them an arbitrarily large distance in the direction −→v without hitting another element of S. We say that Si ≺ Sj if Sj has to be removed before we can remove Si . The relation ≺ defines a partial order P(S, ≺) on S which we call the separability order of S and −→v . A partial order P(X, ≺ ) on X = {x1,..., xn} is called a separability order if there is a collection of convex sets S and a vector −→v in some Rd such that xi ≺ x j in P(X, ≺ ) if and only if Si ≺ Sj in P(S, ≺). We prove that every partial order is the separability order of a collection of convex sets in R4, and that any poset of dimension 2 is the separability order of a set of line segments in R3. We then study the case when the convex sets are restricted to be boxes in d-dimensional spaces. We prove that any partial order is the separability order of a family of disjoint boxes in Rd for some d ≤ n 2 + 1. We prove that every poset of dimension 3 has a subdivision that is the separability order of boxes in R3, that there are partial orders of dimension 2 that cannot be realized as box separability in R3 and that for any d there are posets with dimension d that are separability orders of boxes in R3. We also prove that for any d there are partial orders with box separability dimension d; that is, d is the smallest dimension for which they are separable orders of sets of boxes in Rd