A polynomial kernel for Block Graph Deletion

In the Block Graph Deletion problem, we are given a graph $G$ on $n$ vertices and a positive integer $k$, and the objective is to check whether it is possible to delete at most $k$ vertices from $G$ to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with $\mathcal{O}(k^{6})$ vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of `complete degree' of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time $10^{k}\cdot n^{\mathcal{O}(1)}$.Comment: 22 pages, 2 figures, An extended abstract appeared in IPEC201

An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion

Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514--528, 2006]. Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an $n$-vertex graph $G$ and a positive integer $k$, we want to decide whether there is a set of at most $k$ vertices whose removal turns $G$ into a graph of linear rankwidth at most $1$ and find such a vertex set if one exists. While the meta-theorem of Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved in time $f(k)\cdot n^3$ for some function $f$, it is not clear whether this problem allows a running time with a modest exponential function. We first establish that LRW1-Vertex Deletion can be solved in time $8^k\cdot n^{\mathcal{O}(1)}$. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define necklace graphs and investigate their structural properties. Later, we reduce the polynomial factor by refining the trivial branching step based on a cliquewidth expression of a graph, and obtain an algorithm that runs in time $2^{\mathcal{O}(k)}\cdot n^4$. We also prove that the running time cannot be improved to $2^{o(k)}\cdot n^{\mathcal{O}(1)}$ under the Exponential Time Hypothesis assumption. Lastly, we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.Comment: 29 pages, 9 figures, An extended abstract appeared in IPEC201

Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k

International audienceEvery minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field $\mathbb F$, the list contains only finitely many $\mathbb F$-representable matroids, due to the well-quasi-ordering of $\mathbb F$-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these $\mathbb F$-representable excluded minors in general. We consider the class of matroids of path-width at most $k$ for fixed $k$. We prove that for a finite field $\mathbb F$, every $\mathbb F$-representable excluded minor for the class of matroids of path-width at most~$k$ has at most $2^{|\mathbb{F}|^{O(k^2)}}$ elements. We can therefore compute, for any integer $k$ and a fixed finite field $\mathbb F$, the set of $\mathbb F$-representable excluded minors for the class of matroids of path-width $k$, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an $\mathbb F$-represented matroid is at most $k$. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most $k$ has at most $2^{2^{O(k^2)}}$ vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs

Lung function, coronary artery calcification, and metabolic syndrome in 4905 Korean males

SummaryBackgroundImpaired lung function is an independent predictor of cardiovascular mortality. We assessed the relationships of lung function with insulin resistance (IR), metabolic syndrome (MetS), systemic inflammation and coronary artery calcification score (CACS) measured by computed tomography (CT) scan an indicator of coronary atherosclerosis.MethodsWe identified 4905 adult male patients of the Health Promotion Center in Samsung Medical Center between March 2005 and February 2008 and retrospectively reviewed the following data for these patients: pulmonary function, CT-measured CACS, anthropometric measurement, fasting glucose, insulin, lipid profiles, serum C-reactive protein (CRP) and homeostatic model assessment (HOMA-IR). MetS was defined according to the AHA/NHLBI criteria.ResultsWhen the subjects were divided into four groups according to quartiles of FVC or FEV1 (% pred), serum CRP level, HOMA-IR, prevalence of MetS and CACS significantly increased as the FVC or FEV1 (% pred) decreased. The odds ratios (ORs) for MetS in the lowest quartiles of FVC and FEV1 (% pred) were 1.85 (95% CI, 1.49–2.30; p<0.001) and 1.47 (95% CI, 1.20–1.81; p<0.001) respectively. The ORs for the presence of coronary artery calcification in the lowest quartiles of FVC and FEV1 (% pred) were 1.31 (95% CI, 1.09–1.58; p=0.004) and 1.22 (95% CI, 1.02–1.46; p=0.029) respectively. Obesity, CRP, HOMA-IR, and the presence of coronary artery calcium were independent risk predictors for impaired lung function.ConclusionMetabolic syndrome, insulin resistance, coronary atherosclerosis, and systemic inflammation are closely related to the impaired lung function

Information flow between stock indices

Using transfer entropy, we observed the strength and direction of information flow between stock indices. We uncovered that the biggest source of information flow is America. In contrast, the Asia/Pacific region the biggest is receives the most information. According to the minimum spanning tree, the GSPC is located at the focal point of the information source for world stock markets