Orientations making k-cycles cyclic

We show that the minimum number of orientations of the edges of the n-vertex complete graph having the property that every triangle is made cyclic in at least one of them is $\lceil\log_2(n-1)\rceil$. More generally, we also determine the minimum number of orientations of $K_n$ such that at least one of them orients some specific $k$-cycles cyclically on every $k$-element subset of the vertex set. The questions answered by these results were motivated by an analogous problem of Vera T. S\'os concerning triangles and $3$-edge-colorings. Some variants of the problem are also considered.Comment: 9 page

Tangle-tree duality in abstract separation systems

We prove a general width duality theorem for combinatorial structures with well-defined notions of cohesion and separation. These might be graphs and matroids, but can be much more general or quite different. The theorem asserts a duality between the existence of high cohesiveness somewhere local and a global overall tree structure. We describe cohesive substructures in a unified way in the format of tangles: as orientations of low-order separations satisfying certain consistency axioms. These axioms can be expressed without reference to the underlying structure, such as a graph or matroid, but just in terms of the poset of the separations themselves. This makes it possible to identify tangles, and apply our tangle-tree duality theorem, in very diverse settings. Our result implies all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width or rank-width. It yields new, tangle-type, duality theorems for tree-width and path-width. It implies the existence of width parameters dual to cohesive substructures such as $k$-blocks, edge-tangles, or given subsets of tangles, for which no width duality theorems were previously known. Abstract separation systems can be found also in structures quite unlike graphs and matroids. For example, our theorem can be applied to image analysis by capturing the regions of an image as tangles of separations defined as natural partitions of its set of pixels. It can be applied in big data contexts by capturing clusters as tangles. It can be applied in the social sciences, e.g. by capturing as tangles the few typical mindsets of individuals found by a survey. It could also be applied in pure mathematics, e.g. to separations of compact manifolds.Comment: We have expanded Section 2 on terminology for better readability, adding explanatory text, examples, and figures. This paper replaces the first half of our earlier paper arXiv:1406.379

The power of phylogenetic approaches to detect horizontally transferred genes

BACKGROUND: Horizontal gene transfer plays an important role in evolution because it sometimes allows recipient lineages to adapt to new ecological niches. High genes transfer frequencies were inferred for prokaryotic and early eukaryotic evolution. Does horizontal gene transfer also impact phylogenetic reconstruction of the evolutionary history of genomes and organisms? The answer to this question depends at least in part on the actual gene transfer frequencies and on the ability to weed out transferred genes from further analyses. Are the detected transfers mainly false positives, or are they the tip of an iceberg of many transfer events most of which go undetected by current methods? RESULTS: Phylogenetic detection methods appear to be the method of choice to infer gene transfers, especially for ancient transfers and those followed by orthologous replacement. Here we explore how well some of these methods perform using in silico transfers between the terminal branches of a gamma proteobacterial, genome based phylogeny. For the experiments performed here on average the AU test at a 5% significance level detects 90.3% of the transfers and 91% of the exchanges as significant. Using the Robinson-Foulds distance only 57.7% of the exchanges and 60% of the donations were identified as significant. Analyses using bipartition spectra appeared most successful in our test case. The power of detection was on average 97% using a 70% cut-off and 94.2% with 90% cut-off for identifying conflicting bipartitions, while the rate of false positives was below 4.2% and 2.1% for the two cut-offs, respectively. For all methods the detection rates improved when more intervening branches separated donor and recipient. CONCLUSION: Rates of detected transfers should not be mistaken for the actual transfer rates; most analyses of gene transfers remain anecdotal. The method and significance level to identify potential gene transfer events represent a trade-off between the frequency of erroneous identification (false positives) and the power to detect actual transfer events

Dynamic programming for graphs on surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2O(k·log k). Our approach combines tools from topological graph theory and analytic combinatorics.Postprint (updated version

Geometric Separation and Packing Problems

The first part of this thesis investigates combinatorial and algorithmic aspects of geometric separation problems in the plane. In such a setting one is given a set of points and a set of separators such as lines, line segments or disks. The goal is to select a small subset of those separators such that every path between any two points is intersected by at least one separator. We first look at several problems which arise when one is given a set of points and a set of unit disks embedded in the plane and the goal is to separate the points using a small subset of the given disks. Next, we focus on a separation problem involving only one region: Given a region in the plane, bounded by a piecewise linear closed curve, such as a fence, place few guards inside the fenced region such that wherever an intruder cuts through the fence, the closest guard is at most a distance one away. Restricting the separating objects to be lines, we investigate combinatorial aspects which arise when we use them to pairwise separate a set of points in the plane; hereafter we generalize the notion of separability to arbitrary sets and present several enumeration results. Lastly, we investigate a packing problem with a non-convex shape in ℝ3. We show that ℝ3 can be packed at a density of 0.222 with tori of major radius one and minor radius going to zero. Furthermore, we show that the same torus arrangement yields the asymptotically optimal number of pairwise linked tori