Efficient domination and polarity

The thesis considers the following graph problems: Efficient (Edge) Domination seeks for an independent vertex (edge) subset D such that all other vertices (edges) have exactly one neighbor in D. Polarity asks for a vertex subset that induces a complete multipartite graph and that contains a vertex of every induced P_3. Monopolarity is the special case of Polarity where the wanted vertex subset has to be independent. These problems are NP-complete in general, but efficiently solvable on various graph classes. The thesis sharpens known NP-completeness results and presents new solvable cases

Partitioning a graph into disjoint cliques and a triangle-free graph

A graph G=(V,E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques (i.e., G[A] is P_3-free) and B induces a triangle-free graph (i.e., G[B] is K_3-free). In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be NP-complete on arbitrary graphs. Here it is proved that if a graph G is bull-free, planar, perfect, K_4-free or does not contain certain holes then deciding whether G is partitionable is NP-complete. This answers an open question posed by Thomassé, Trotignon and Vušković. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs

Polar permutation graphs are polynomial-time recognisable *

Abstract Polar graphs generalise bipartite graphs, cobipartite graphs, and split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete problem. Here, we show that for permutation graphs this problem can be solved in polynomial time. The result is surprising, as related problems like achromatic number and cochromatic number are NP-complete on permutation graphs. We give a polynomial-time algorithm for recognising graphs that are both permutation and polar. Prior to our result, polarity has been resolved only for chordal graphs and cographs

Condensation of the fusion focus by the intrinsically disordered region of the formin Fus1 is essential for cell-cell fusion.

Secretory vesicle clusters transported on actin filaments by myosin V motors for local secretion underlie various cellular processes, such as neurotransmitter release at neuronal synapses, <sup>1</sup> hyphal steering in filamentous fungi, <sup>2</sup> <sup>,</sup> <sup>3</sup> and local cell wall digestion preceding the fusion of yeast gametes. <sup>4</sup> During fission yeast Schizosaccharomyces pombe gamete fusion, the actin fusion focus assembled by the formin Fus1 concentrates secretory vesicles carrying cell wall digestive enzymes. <sup>5-7</sup> The position and coalescence of the vesicle focus are controlled by local signaling and actin-binding proteins to prevent inappropriate cell wall digestion that would cause lysis, <sup>6</sup> <sup>,</sup> <sup>8-10</sup> but the mechanisms of focusing have been elusive. Here, we show that the regulatory N terminus of Fus1 contains an intrinsically disordered region (IDR) that mediates Fus1 condensation in vivo and forms dense assemblies that exclude ribosomes. Fus1 lacking its IDR fails to concentrate in a tight focus and causes cell lysis during attempted cell fusion. Remarkably, the replacement of Fus1 IDR with a heterologous low-complexity region that forms molecular condensates fully restores Fus1 focusing and function. By contrast, the replacement of Fus1 IDR with a domain that forms more stable oligomers restores focusing but poorly supports cell fusion, suggesting that condensation is tuned to yield a selectively permeable structure. We propose that condensation of actin structures by an IDR may be a general mechanism for actin network organization and the selective local concentration of secretory vesicles

Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

A fundamental graph problem is to recognize whether the vertex set of a graph G can be bipartitioned into sets A and B such that G[A] and G[B] satisfy properties Pi_A and Pi_B, respectively. This so-called (Pi_A,Pi_B)-Recognition problem generalizes amongst others the recognition of 3-colorable, bipartite, split, and monopolar graphs. A powerful algorithmic technique that can be used to obtain fixed-parameter algorithms for many cases of (Pi_A,Pi_B)-Recognition, as well as several other problems, is the pushing process. For bipartition problems, the process starts with an "almost correct" bipartition (A\u27,B\u27), and pushes appropriate vertices from A\u27 to B\u27 and vice versa to eventually arrive at a correct bipartition. In this paper, we study whether (Pi_A,Pi_B)-Recognition problems for which the pushing process yields fixed-parameter algorithms also admit polynomial problem kernels. In our study, we focus on the first level above triviality, where Pi_A is the set of P_3-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph G[A], and Pi_B is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that NP not subseteq coNP/poly, (Pi_A,Pi_B)-Recognition admits a polynomial kernel if and only if H contains a graph of order at most 2. In both the kernelization and the lower bound results, we make crucial use of the pushing process

Investigation into the role of Aurora A kinase activity during mitosis

Aurora A is an important mitotic regulator that has been found to be up-regulated in a variety oftumours provoking a great deal of attention and the development of a number of small moleculeAurora kinase inhibitors. Most of these inhibitors though have predominantly targeted Aurora B,meaning that our understanding of the role of the kinase activity of Aurora A is comparatively lesswell developed.MLN8054 however, is a small molecule inhibitor that has been reported in vitro to have a highdegree of specificity towards Aurora A activity. In this thesis, I show in vivo that MLN8054 can beused to specifically inhibit Aurora A activity, and exploit this quality to probe the role of Aurora Aactivity in human cells. I was consequently able to show that Aurora A activity not only has a clearrole in spindle formation, where it is required for the determination of K-fibre length and in thedegree of centrosome separation, but also in the regulation of microtubule organisation. Despite thespindle deformities seen after inhibiting Aurora A activity, the majority of HeLa and DLD-1 cellswere still able to form bipolar spindles capable of attaching to kinetochores. These spindlestructures did not however, assert normal levels of force through the kinetochores, and cells wereconsequently unable to efficiently align their chromosomes, causing significant delays to mitoticprogression. Cells were still able to divide in the absence of Aurora A activity, although thedetection of segregation defects and aneuploid progeny indicates a role for Aurora A activity in thefaithful segregation of the genetic material. Importantly however, Aurora A activity was not foundto have a prominent role in the spindle assembly checkpoint.Increasing the potency of Aurora A inhibition by using a drug-resistant cell line confirmed theobservations made in HeLa and DLD-1 cells, emphasising that although Aurora A activity isrequired for spindle assembly, cells can still activate the spindle checkpoint and divide in itsabsence. I therefore propose that Aurora A activity is required for the formation of normal spindlestructures capable of efficiently aligning and evenly dividing chromosomes during cell division.These roles were attributed in part to the kinase activity of Aurora A in the regulation of TACC3and chTOG localisation on the spindle and centrosomes.Interestingly however, Aurora A activity did not appear to be required for spindle assembly in nontransformedcells, which were able to more efficiently align their chromosomes and dividefollowing Aurora A inhibition than the cancer cell lines. Furthermore, the non-transformed cellsaccumulated with 2N DNA after longer-term Aurora A inhibition, as opposed to the cancer celllines, which exhibited profound aneuploidy following the equivalent treatment. This finding isencouraging, as consistent with recently published reports, it indicates that Aurora A inhibitionmay be successfully used in order to specifically target cancer cells.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Topics in graph colouring and extremal graph theory

In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let $G$ be a connected graph with $n$ vertices and maximum degree $\Delta(G)$. Let $R_k(G)$ denote the graph with vertex set all proper $k$-colourings of $G$ and two $k$-colourings are joined by an edge if they differ on the colour of exactly one vertex. Our first main result states that $R_{\Delta(G)+1}(G)$ has a unique non-trivial component with diameter $O(n^2)$. This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree. A Kempe change is the operation of swapping some colours $a$, $b$ of a component of the subgraph induced by vertices with colour $a$ or $b$. Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all $\Delta(G)$-colourings of a graph $G$ are Kempe equivalent unless $G$ is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007). Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs. Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees

Models for spatial organization of microtubules and cell polarization

The main actors in this PhD thesis are microtubules, dynamic polymers built from protein subunits that play many important roles in cells of all higher organisms. We study a number of their functions using mathematical modelling and computer simulations. First, we consider the role of microtubules in establishing the ordered cortical array, a structure that plays a key role in the plant cell elongation. We show theoretically that the observed effect that new microtubules are nucleated from pre-existing microtubules strongly enhances the order of the array and makes it more robust against variations in the conditions. Next, we turn to the role the shape of the cell plays in the spatial organization of microtubules. We show that depending on how the microtubules interact with the cell boundary, either the long or the short axis of the cell determines the majority direction of the microtubules. Additionally, we formulate a model that predicts the positioning of the mitotic spindle, which is the cellular structure that segregates the duplicated chromosomes during eukaryotic cell division. We analyze how the speed, precision and final direction of the spindle orientation process depends on cell shape and the parameters that describe the microtubules. Finally, we turn to the potential role of microtubules in establishing cell polarity, the non-uniform distribution of cellular constituents, which is crucial for many developmental processes. Based solely on the propensity of microtubules to bind and transport proteins to the cell membrane, we set up a feasible and robust cell polarization mechanism, which could potentially be used to set up polarity in a minimal cell-like environment using a biochemical reconstitution approach. We study this model in its pure form in a perfectly spherical cell, in order to establish proof-of-principle, but also show that the effect remains in a more realistic non-spherical cell.</p