An FPT 2-Approximation for Tree-Cut Decomposition

The tree-cut width of a graph is a graph parameter defined by Wollan [J. Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut decompositions. In certain cases, tree-cut width appears to be more adequate than treewidth as an invariant that, when bounded, can accelerate the resolution of intractable problems. While designing algorithms for problems with bounded tree-cut width, it is important to have a parametrically tractable way to compute the exact value of this parameter or, at least, some constant approximation of it. In this paper we give a parameterized 2-approximation algorithm for the computation of tree-cut width; for an input $n$-vertex graph $G$ and an integer $w$, our algorithm either confirms that the tree-cut width of $G$ is more than $w$ or returns a tree-cut decomposition of $G$ certifying that its tree-cut width is at most $2w$, in time $2^{O(w^2\log w)} \cdot n^2$. Prior to this work, no constructive parameterized algorithms, even approximated ones, existed for computing the tree-cut width of a graph. As a consequence of the Graph Minors series by Robertson and Seymour, only the existence of a decision algorithm was known.Comment: 17 pages, 3 figure

Temperate southern Australian coastal waters are characterised by surprisingly high rates of nitrogen fixation and diversity of diazotrophs

Biological dinitrogen (N2) fixation is one mechanism by which specific microorganisms (diazotrophs) can ameliorate nitrogen (N) limitation. Historically, rates of N2 fixation were believed to be limited outside of the low nutrient tropical and subtropical open ocean; however, emerging evidence suggests that N2 fixation is also a significant process within temperate coastal waters. Using a combination of amplicon sequencing, targeting the nitrogenase reductase gene (nifH), quantitative nifH PCR, and 15N2 stable isotope tracer experiments, we investigated spatial patterns of diazotroph assemblage structure and N2 fixation rates within the temperate coastal waters of southern Australia during Austral autumn and summer. Relative to previous studies in open ocean environments, including tropical northern Australia, and tropical and temperate estuaries, our results indicate that high rates of N2 fixation (10–64 nmol L−1 d−1) can occur within the large inverse estuary Spencer Gulf, while comparatively low rates of N2 fixation (2 nmol L−1 d−1) were observed in the adjacent continental shelf waters. Across the dataset, low concentrations of NO3/NO2 were significantly correlated with the highest N2 fixation rates, suggesting that N2 fixation could be an important source of new N in the region as dissolved inorganic N concentrations are typically limiting. Overall, the underlying diazotrophic community was dominated by nifH sequences from Cluster 1 unicellular cyanobacteria of the UCYN-A clade, as well as non-cyanobacterial diazotrophs related to Pseudomonas stutzeri, and Cluster 3 sulfate-reducing deltaproteobacteria. Diazotroph community composition was significantly influenced by salinity and SiO4 concentrations, reflecting the transition from UCYN-A-dominated assemblages in the continental shelf waters, to Cluster 3-dominated assemblages in the hypersaline waters of the inverse estuary. Diverse, transitional diazotrophic communities, comprised of a mixture of UCYN-A and putative heterotrophic bacteria, were observed at the mouth and southern edge of Spencer Gulf, where the highest N2 fixation rates were observed. In contrast to observations in other environments, no seasonal patterns in N2 fixation rates and diazotroph community structure were apparent. Collectively, our findings are consistent with the emerging view that N2 fixation within temperate coastal waters is a previously overlooked dynamic and potentially important component of the marine N cycle

Idealness of k-wise intersecting families

A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that, for some integer k ≥ 4, every k-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for k = 4 for the class of binary clutters. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. We also discuss connections to the chromatic number of a clutter, projective geometries over the two-element field, uniform cycle covers in graphs, and quarter-integral packings of value two in ideal clutters

On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.Comment: 20

The Effect of Planarization on Width

We study the effects of planarization (the construction of a planar diagram $D$ from a non-planar graph $G$ by replacing each crossing by a new vertex) on graph width parameters. We show that for treewidth, pathwidth, branchwidth, clique-width, and tree-depth there exists a family of $n$-vertex graphs with bounded parameter value, all of whose planarizations have parameter value $\Omega(n)$. However, for bandwidth, cutwidth, and carving width, every graph with bounded parameter value has a planarization of linear size whose parameter value remains bounded. The same is true for the treewidth, pathwidth, and branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Parameterized Directed kk-Chinese Postman Problem and kk Arc-Disjoint Cycles Problem on Euler Digraphs

In the Directed $k$-Chinese Postman Problem ($k$-DCPP), we are given a connected weighted digraph $G$ and asked to find $k$ non-empty closed directed walks covering all arcs of $G$ such that the total weight of the walks is minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128) asked for the parameterized complexity of $k$-DCPP when $k$ is the parameter. We prove that the $k$-DCPP is fixed-parameter tractable. We also consider a related problem of finding $k$ arc-disjoint directed cycles in an Euler digraph, parameterized by $k$. Slivkins (ESA 2003) showed that this problem is W[1]-hard for general digraphs. Generalizing another result by Slivkins, we prove that the problem is fixed-parameter tractable for Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler digraphs remains W[1]-hard even for Euler digraphs

Nonlinearization and waves in bounded media: old wine in a new bottle

We consider problems such as a standing wave in a closed straight tube, a self-sustained oscillation, damped resonance, evolution of resonance and resonance between concentric spheres. These nonlinear problems, and other similar ones, have been solved by a variety of techniques when it is seen that linear theory fails. The unifying approach given here is to initially set up the appropriate linear difference equation, where the difference is the linear travel time. When the linear travel time is replaced by a corrected nonlinear travel time, the nonlinear difference equation yields the required solution

Identically self-blocking clutters

A clutter is identically self-blocking if it is equal to its blocker. We prove that every identically self-blocking clutter different from is nonideal. Our proofs borrow tools from Gauge Duality and Quadratic Programming. Along the way we provide a new lower bound for the packing number of an arbitrary clutter

Network inference analysis identifies an APRR2-like gene linked to pigment accumulation in tomato and pepper fruits

Carotenoids represent some of the most important secondary metabolites in the human diet, and tomato (Solanum lycopersicum) is a rich source of these health-promoting compounds. In this work, a novel and fruit-related regulator of pigment accumulation in tomato has been identified by artificial neural network inference analysis and its function validated in transgenic plants. A tomato fruit gene regulatory network was generated using artificial neural network inference analysis and transcription factor gene expression profiles derived from fruits sampled at various points during development and ripening. One of the transcription factor gene expression profiles with a sequence related to an Arabidopsis (Arabidopsis thaliana) ARABIDOPSIS PSEUDO RESPONSE REGULATOR2-LIKE gene (APRR2-Like) was up-regulated at the breaker stage in wild-type tomato fruits and, when overexpressed in transgenic lines, increased plastid number, area, and pigment content, enhancing the levels of chlorophyll in immature unripe fruits and carotenoids in red ripe fruits. Analysis of the transcriptome of transgenic lines overexpressing the tomato APPR2-Like gene revealed up-regulation of several ripening-related genes in the overexpression lines, providing a link between the expression of this tomato gene and the ripening process. A putative ortholog of the tomato APPR2-Like gene in sweet pepper (Capsicum annuum) was associated with pigment accumulation in fruit tissues. We conclude that the function of this gene is conserved across taxa and that it encodes a protein that has an important role in ripening

Bond percolation on isoradial graphs: criticality and universality

In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star-triangle transformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and 2j-alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex