Graph Treewidth and Geometric Thickness Parameters

Consider a drawing of a graph $G$ in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of $G$, is the classical graph parameter "thickness". By restricting the edges to be straight, we obtain the "geometric thickness". By further restricting the vertices to be in convex position, we obtain the "book thickness". This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth $k$, the maximum thickness and the maximum geometric thickness both equal $\lceil{k/2}\rceil$. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth $k$, the maximum book thickness equals $k$ if $k \leq 2$ and equals $k+1$ if $k \geq 3$. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in Computer Science 3843:129-140, Springer, 2006. The full version was published in Discrete & Computational Geometry 37(4):641-670, 2007. That version contained a false conjecture, which is corrected on page 26 of this versio

Graph Treewidth and Geometric Thickness Parameters

Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity

Stoma-free survival after anastomotic leak following rectal cancer resection: worldwide cohort of 2470 patients

Background: The optimal treatment of anastomotic leak after rectal cancer resection is unclear. This worldwide cohort study aimed to provide an overview of four treatment strategies applied. Methods: Patients from 216 centres and 45 countries with anastomotic leak after rectal cancer resection between 2014 and 2018 were included. Treatment was categorized as salvage surgery, faecal diversion with passive or active (vacuum) drainage, and no primary/secondary faecal diversion. The primary outcome was 1-year stoma-free survival. In addition, passive and active drainage were compared using propensity score matching (2: 1). Results: Of 2470 evaluable patients, 388 (16.0 per cent) underwent salvage surgery, 1524 (62.0 per cent) passive drainage, 278 (11.0 per cent) active drainage, and 280 (11.0 per cent) had no faecal diversion. One-year stoma-free survival rates were 13.7, 48.3, 48.2, and 65.4 per cent respectively. Propensity score matching resulted in 556 patients with passive and 278 with active drainage. There was no statistically significant difference between these groups in 1-year stoma-free survival (OR 0.95, 95 per cent c.i. 0.66 to 1.33), with a risk difference of -1.1 (95 per cent c.i. -9.0 to 7.0) per cent. After active drainage, more patients required secondary salvage surgery (OR 2.32, 1.49 to 3.59), prolonged hospital admission (an additional 6 (95 per cent c.i. 2 to 10) days), and ICU admission (OR 1.41, 1.02 to 1.94). Mean duration of leak healing did not differ significantly (an additional 12 (-28 to 52) days). Conclusion: Primary salvage surgery or omission of faecal diversion likely correspond to the most severe and least severe leaks respectively. In patients with diverted leaks, stoma-free survival did not differ statistically between passive and active drainage, although the increased risk of secondary salvage surgery and ICU admission suggests residual confounding

Direct observation of the dead-cone effect in quantum chromodynamics

At particle collider experiments, elementary particle interactions with large momentum transfer produce quarks and gluons (known as partons) whose evolution is governed by the strong force, as described by the theory of quantum chromodynamics (QCD) [1]. The vacuum is not transparent to the partons and induces gluon radiation and quark pair production in a process that can be described as a parton shower [2]. Studying the pattern of the parton shower is one of the key experimental tools in understanding the properties of QCD. This pattern is expected to depend on the mass of the initiating parton, through a phenomenon known as the dead-cone effect, which predicts a suppression of the gluon spectrum emitted by a heavy quark of mass m and energy E, within a cone of angular size m/E around the emitter [3]. A direct observation of the dead-cone effect in QCD has not been possible until now, due to the challenge of reconstructing the cascading quarks and gluons from the experimentally accessible bound hadronic states. Here we show the first direct observation of the QCD dead-cone by using new iterative declustering techniques [4, 5] to reconstruct the parton shower of charm quarks. This result confirms a fundamental feature of QCD, which is derived more generally from its origin as a gauge quantum field theory. Furthermore, the measurement of a dead-cone angle constitutes the first direct experimental observation of the non-zero mass of the charm quark, which is a fundamental constant in the standard model of particle physics.The direct measurement of the QCD dead cone in charm quark fragmentation is reported, using iterative declustering of jets tagged with a fully reconstructed charmed hadron.In particle collider experiments, elementary particle interactions with large momentum transfer produce quarks and gluons (known as partons) whose evolution is governed by the strong force, as described by the theory of quantum chromodynamics (QCD). These partons subsequently emit further partons in a process that can be described as a parton shower which culminates in the formation of detectable hadrons. Studying the pattern of the parton shower is one of the key experimental tools for testing QCD. This pattern is expected to depend on the mass of the initiating parton, through a phenomenon known as the dead-cone effect, which predicts a suppression of the gluon spectrum emitted by a heavy quark of mass $m_{\rm{Q}}$ and energy $E$, within a cone of angular size $m_{\rm{Q}}$/$E$ around the emitter. Previously, a direct observation of the dead-cone effect in QCD had not been possible, owing to the challenge of reconstructing the cascading quarks and gluons from the experimentally accessible hadrons. We report the direct observation of the QCD dead cone by using new iterative declustering techniques to reconstruct the parton shower of charm quarks. This result confirms a fundamental feature of QCD. Furthermore, the measurement of a dead-cone angle constitutes a direct experimental observation of the non-zero mass of the charm quark, which is a fundamental constant in the standard model of particle physics