On the threshold-width of graphs

The GG-width of a class of graphs GG is defined as follows. A graph G has GG-width k if there are k independent sets N1,...,Nk in G such that G can be embedded into a graph H in GG such that for every edge e in H which is not an edge in G, there exists an i such that both endpoints of e are in Ni. For the class TH of threshold graphs we show that TH-width is NP-complete and we present fixed-parameter algorithms. We also show that for each k, graphs of TH-width at most k are characterized by a finite collection of forbidden induced subgraphs

Subgraphs and Colourability of Locatable Graphs

We study a game of pursuit and evasion introduced by Seager in 2012, in which a cop searches the robber from outside the graph, using distance queries. A graph on which the cop wins is called locatable. In her original paper, Seager asked whether there exists a characterisation of the graph property of locatable graphs by either forbidden or forbidden induced subgraphs, both of which we answer in the negative. We then proceed to show that such a characterisation does exist for graphs of diameter at most 2, stating it explicitly, and note that this is not true for higher diameter. Exploring a different direction of topic, we also start research in the direction of colourability of locatable graphs, we also show that every locatable graph is 4-colourable, but not necessarily 3-colourable.Comment: 25 page

Probing the (H3-H4)(2) histone tetramer structure using pulsed EPR spectroscopy combined with site-directed spin labelling

The (H3-H4)2 histone tetramer forms the central core of nucleosomes and, as such, plays a prominent role in assembly, disassembly and positioning of nucleosomes. Despite its fundamental role in chromatin, the tetramer has received little structural investigation. Here, through the use of pulsed electron-electron double resonance spectroscopy coupled with site-directed spin labelling, we survey the structure of the tetramer in solution. We find that tetramer is structurally more heterogeneous on its own than when sequestered in the octamer or nucleosome. In particular, while the central region including the H3-H3′ interface retains a structure similar to that observed in nucleosomes, other regions such as the H3 αN helix display increased structural heterogeneity. Flexibility of the H3 αN helix in the free tetramer also illustrates the potential for post-translational modifications to alter the structure of this region and mediate interactions with histone chaperones. The approach described here promises to prove a powerful system for investigating the structure of additional assemblies of histones with other important factors in chromatin assembly/fluidity

Loss of strumpellin in the melanocytic lineage impairs the WASH Complex but does not affect coat colour

The five-subunit WASH complex generates actin networks that participate in endocytic trafficking, migration and invasion in various cell types. Loss of one of the two subunits WASH or strumpellin in mice is lethal, but little is known about their role in mammals in vivo. We explored the role of strumpellin, which has previously been linked to hereditary spastic paraplegia, in the mouse melanocytic lineage. Strumpellin knockout in melanocytes revealed abnormal endocytic vesicle morphology but no impairment of migration in vitro or in vivo and no change in coat colour. Unexpectedly, WASH and filamentous actin could still localize to vesicles in the absence of strumpellin, although the shape and size of vesicles was altered. Blue native PAGE revealed the presence of two distinct WASH complexes, even in strumpellin knockout cells, revealing that the WASH complex can assemble and localize to endocytic compartments in cells in the absence of strumpellin

Dynamic representation of consecutive-ones matrices and interval graphs

2015 Spring.Includes bibliographical references.We give an algorithm for updating a consecutive-ones ordering of a consecutive-ones matrix when a row or column is added or deleted. When the addition of the row or column would result in a matrix that does not have the consecutive-ones property, we return a well-known minimal forbidden submatrix for the consecutive-ones property, known as a Tucker submatrix, which serves as a certificate of correctness of the output in this case, in O(n log n) time. The ability to return such a certificate within this time bound is one of the new contributions of this work. Using this result, we obtain an O(n) algorithm for updating an interval model of an interval graph when an edge or vertex is added or deleted. This matches the bounds obtained by a previous dynamic interval-graph recognition algorithm due to Crespelle. We improve on Crespelle's result by producing an easy-to-check certificate, known as a Lekkerkerker-Boland subgraph, when a proposed change to the graph results in a graph that is not an interval graph. Our algorithm takes O(n log n) time to produce this certificate. The ability to return such a certificate within this time bound is the second main contribution of this work

Penrose limits, supergravity and brane dynamics

We investigate the Penrose limits of classical string and M-theory backgrounds. We prove that the number of (super)symmetries of a supergravity background never decreases in the limit. We classify all the possible Penrose limits of AdS x S spacetimes and of supergravity brane solutions. We also present the Penrose limits of various other solutions: intersecting branes, supersymmetric black holes and strings in diverse dimensions, and cosmological models. We explore the Penrose limit of an isometrically embedded spacetime and find a generalisation to spaces with more than one time. Finally, we show that the Penrose limit is a large tension limit for all branes including those with fields of Born--Infeld type.Comment: 67 page

Recognition Algorithm for Probe Interval 2-Trees

Recognition of probe interval graphs has been studied extensively. Recognition algorithms of probe interval graphs can be broken down into two types of problems: partitioned and non-partitioned. A partitioned recognition algorithm includes the probe and nonprobe partition of the vertices as part of the input, where a non-partitioned algorithm does not include the partition. Partitioned probe interval graphs can be recognized in linear-time in the edges, whereas non-partitioned probe interval graphs can be recognized in polynomial-time. Here we present a non-partitioned recognition algorithm for 2-trees, an extension of trees, that are probe interval graphs. We show that this algorithm runs in O(m) time, where m is the number of edges of a 2-tree. Currently there is no algorithm that performs as well for this problem