2-Vertex Connectivity in Directed Graphs

We complement our study of 2-connectivity in directed graphs, by considering the computation of the following 2-vertex-connectivity relations: We say that two vertices v and w are 2-vertex-connected if there are two internally vertex-disjoint paths from v to w and two internally vertex-disjoint paths from w to v. We also say that v and w are vertex-resilient if the removal of any vertex different from v and w leaves v and w in the same strongly connected component. We show how to compute the above relations in linear time so that we can report in constant time if two vertices are 2-vertex-connected or if they are vertex-resilient. We also show how to compute in linear time a sparse certificate for these relations, i.e., a subgraph of the input graph that has O(n) edges and maintains the same 2-vertex-connectivity and vertex-resilience relations as the input graph, where n is the number of vertices.Comment: arXiv admin note: substantial text overlap with arXiv:1407.304

Status, Trends, and Conservation of Eelgrass in Atlantic Canada and the Northeastern United States: Workshop Report

Eelgrass (Zostera marina L) is the dominant seagrass occurring in eastern Canada and the northeastern United States, where it often forms extensive meadows in coastal and estuarine areas. Eelgrass beds are extremely productive and provide many valuable ecological functions and ecosystem services. They serve as critical feeding and nursery habitat for a wide variety of commercially and recreationally important fish and shellfish and as feeding areas for waterfowl and other waterbirds. Eelgrass detritus is also transported considerable distances to fuel offshore food webs. In addition, eelgrass beds stabilize bottom sediments, dampen wave energy, absorb nutrients from surrounding waters, and retain carbon through burial

Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time

We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple $O(m n)$-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time $O(n^2)$. For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an $O(m^2 / \log{n})$-time algorithm for 2-edge strongly connected components, and thus improve over the $O(m n)$ running time also when $m = O(n)$. Our approach extends to k-edge and k-vertex strongly connected components for any constant k with a running time of $O(n^2 \log^2 n)$ for edges and $O(n^3)$ for vertices

Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count

We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter $D=\Omega(\sqrt n)$, and at most $\min\bigl(2n-\Omega(\sqrt n),2n-D-2\bigr)$ edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

An Improved Upper Bound for the Ring Loading Problem

The Ring Loading Problem emerged in the 1990s to model an important special case of telecommunication networks (SONET rings) which gained attention from practitioners and theorists alike. Given an undirected cycle on $n$ nodes together with non-negative demands between any pair of nodes, the Ring Loading Problem asks for an unsplittable routing of the demands such that the maximum cumulated demand on any edge is minimized. Let $L$ be the value of such a solution. In the relaxed version of the problem, each demand can be split into two parts where the first part is routed clockwise while the second part is routed counter-clockwise. Denote with $L^*$ the maximum load of a minimum split routing solution. In a landmark paper, Schrijver, Seymour and Winkler [SSW98] showed that $L \leq L^* + 1.5D$, where $D$ is the maximum demand value. They also found (implicitly) an instance of the Ring Loading Problem with $L = L^* + 1.01D$. Recently, Skutella [Sku16] improved these bounds by showing that $L \leq L^* + \frac{19}{14}D$, and there exists an instance with $L = L^* + 1.1D$. We contribute to this line of research by showing that $L \leq L^* + 1.3D$. We also take a first step towards lower and upper bounds for small instances

Containment and equivalence of weighted automata: Probabilistic and max-plus cases

This paper surveys some results regarding decision problems for probabilistic and max-plus automata, such as containment and equivalence. Probabilistic and max-plus automata are part of the general family of weighted automata, whose semantics are maps from words to real values. Given two weighted automata, the equivalence problem asks whether their semantics are the same, and the containment problem whether one is point-wise smaller than the other one. These problems have been studied intensively and this paper will review some techniques used to show (un)decidability and state a list of open questions that still remain

Neural Decision Boundaries for Maximal Information Transmission

We consider here how to separate multidimensional signals into two categories, such that the binary decision transmits the maximum possible information transmitted about those signals. Our motivation comes from the nervous system, where neurons process multidimensional signals into a binary sequence of responses (spikes). In a small noise limit, we derive a general equation for the decision boundary that locally relates its curvature to the probability distribution of inputs. We show that for Gaussian inputs the optimal boundaries are planar, but for non-Gaussian inputs the curvature is nonzero. As an example, we consider exponentially distributed inputs, which are known to approximate a variety of signals from natural environment.Comment: 5 pages, 3 figure

Pixel and Voxel Representations of Graphs

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for $k$-outerplanar graphs with $n$ vertices, $\Theta(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $\Theta(n^2)$ voxels are always sufficient and sometimes necessary for any $n$-vertex graph. We improve this bound to $\Theta(n\cdot \tau)$ for graphs of treewidth $\tau$ and to $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels

The use of porcine small intestinal submucosa mesh (SURGISIS) as a pelvic sling in a man and a woman with previous pelvic surgery: two case reports

<p>Abstract</p> <p>Introduction</p> <p>Closing the pelvic peritoneum to prevent the small bowel dropping into the pelvis after surgery for locally recurrent rectal cancer is important to prevent adhesions deep in the pelvis or complications of adjuvant radiotherapy. Achieving this could be difficult because sufficient native tissue is unavailable; we report on the use of small intestine submucosa extra-cellular matrix mesh in the obliteration of the pelvic brim.</p> <p>Case presentation</p> <p>We describe two cases in which submucosa extra-cellular matrix mesh was used to obliterate the pelvic brim following resection of a recurrent rectal tumour; the first patient, a 78-year-old Caucasian man, presented with small bowel obstruction caused by adhesions to a recurrent rectal tumour. The second patient, an 84-year-old Caucasian woman, presented with vaginal discharge caused by an entero-vaginal fistula due to a recurrent rectal tumour.</p> <p>Conclusion</p> <p>We report on the use of submucosa extra-cellular matrix mesh as a pelvic sling in cases where primary closure of the pelvic peritoneum is unfeasible. Its use had no infective complications and added minimal morbidity to the postoperative period. This is an original case report that would be of interest to general and colorectal surgeons.</p

Human Organ Culture: Updating the Approach to Bridge the Gap from In Vitro to In Vivo in Inflammation, Cancer, and Stem Cell Biology

Human studies, critical for developing new diagnostics and therapeutics, are limited by ethical and logistical issues, and preclinical animal studies are often poor predictors of human responses. Standard human cell cultures can address some of these concerns but the absence of the normal tissue microenvironment can alter cellular responses. Three-dimensional cultures that position cells on synthetic matrices, or organoid or organ-on-a-chip cultures, in which different cell spontaneously organize contacts with other cells and natural matrix only partly overcome this limitation. Here, we review how human organ cultures (HOCs) can more faithfully preserve in vivo tissue architecture and can better represent disease-associated changes. We will specifically describe how HOCs can be combined with both traditional and more modern morphological techniques to reveal how anatomic location can alter cellular responses at a molecular level and permit comparisons among different cells and different cell types within the same tissue. Examples are provided involving use of HOCs to study inflammation, cancer, and stem cell biology.The authors would like to express their gratitude to The National Institute for Health Research (NIHR) Cambridge Biomedical Research Centre (RA-L, JB)