Nonrepetitive colourings of planar graphs with O(log n) colours

A vertex colouring of a graph is nonrepetitive if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The nonrepetitive chromatic number of a graph G is the minimum integer k such that G has a nonrepetitive k-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is O(n−−√) for n-vertex planar graphs. We prove a O(logn) upper bound

Every collinear set in a planar graph is free

We show that if a planar graph G has a plane straight-line drawing in which a subset S of its vertices are collinear, then for any set of points, X, in the plane with |X| = |S|, there is a plane straight-line drawing of G in which the vertices in S are mapped to the points in X. This solves an open problem posed by Ravsky and Verbitsky in 2008. In their terminology, we show that every collinear set is free. This result has applications in graph drawing, including untangling, column planarity, universal point subsets, and partial simultaneous drawings

Every Collinear Set in a Planar Graph is Free

We show that if a planar graph G has a plane straight-line drawing in which a subset S of its vertices are collinear, then for any set of points, X, in the plane with| X| = | S| , there is a plane straight-line drawing of G in which the vertices in S are mapped to the points in X. This solves an open problem posed by Ravsky and Verbitsky (in: Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, arXiv:0806.0253). In their terminology, we show that every collinear set is free. This result has applications in graph drawing, including untangling, column planarity, universal point subsets, and partial simultaneous drawings

Compatible connectivity-augmentation of planar disconnected graphs

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Inflammatory markers and risk factors for recurrence in 35 patients with a post-traumatic chronic subdural haematoma: a prospective study.

OBJECT: To evaluate the role of local inflammation in the pathogenesis and postoperative recurrence of chronic subdural hematoma (CSDH), the authors conducted an investigation in a selected group of patients who could clearly recall a traumatic event and who did not have other risk factors for CSDH. Inflammation was analyzed by measuring the concentration of the proinflammatory and inflammatory cytokines interleukin (IL)-6 and IL-8. The authors also investigated the possible relationship between high levels of local inflammation that were measured and recurrence of the CSDH. METHODS: A prospective study was performed between 1999 and 2001. Thirty-five patients who could clearly recall a traumatic event that had occurred at least 3 weeks previously and who did not have risk factors for CSDH were enrolled. All patients were surgically treated by burr hole irrigation plus external drainage. The concentration of inflammatory cytokines was very high in the lesion, whereas it was normal in serum. In five cases in which recurrence occurred, concentrations of both IL-6 and IL-8 were significantly increased (p < 0.01) in comparison with cases without a recurrence. In a layering hematoma, the IL-6 and IL-8 concentrations were significantly higher (p < 0.05). Layering CSDHs were also significantly correlated with recurrence. Trabecular hematoma had the lowest cytokine levels and the longest median interval between trauma and clinical onset. The interval from trauma did not significantly influence recurrence, although it did differ significantly between the trabecular and layering CSDH groups. Concentrations of IL-6 and IL-8 in the CSDHs did not differ significantly in relation to either the age of the hematoma (measured as the interval from trauma) or the age of the patient. CONCLUSIONS: Brain trauma causes the onset of an inflammatory process within the dural border cell layer; high levels of inflammatory cytokines were significantly correlated with recurrence and layering CSDH. A prolonged postoperative antiinflammatory medicine given as prophylaxis may help prevent the recurrence of a CSDH

Pole dancing: 3D morphs for tree drawings

We study the question whether a crossing-free 3D morph between two straight-line drawings of an n-vertex tree can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with O(rpw(T)) ⊆ O(log n) steps, while for where rpw(T) is the rooted pathwidth or Strahler number of T, while for the latter setting Θ(n) steps are always sufficient and sometimes necessary

Überlegungen zur geplanten Anzeigepflicht bei sexuellem Missbrauch

We introduce the family of k-gap-planar graphs for k≥0, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most k of its crossings. This definition finds motivation in edge casing, as a k-gap-planar graph can be drawn crossing-free after introducing at most k local gaps per edge. We obtain results on the maximum density, drawability of complete graphs, complexity of the recognition problem, and relationships with other families of beyond-planar graphs