On the Complexity of Role Colouring Planar Graphs, Trees and Cographs

We prove several results about the complexity of the role colouring problem. A role colouring of a graph $G$ is an assignment of colours to the vertices of $G$ such that two vertices of the same colour have identical sets of colours in their neighbourhoods. We show that the problem of finding a role colouring with $1< k <n$ colours is NP-hard for planar graphs. We show that restricting the problem to trees yields a polynomially solvable case, as long as $k$ is either constant or has a constant difference with $n$, the number of vertices in the tree. Finally, we prove that cographs are always $k$-role-colourable for $1<k\leq n$ and construct such a colouring in polynomial time

Breaking the Silence: Examining Personal Preparedness of Supporting Students Living with HIV/AIDS

Often, when an article is written, it implies a certain level of expertise on the part of the author. I do not know if one can ever be an expert on supporting friends, family members, and students living with or affected by HIV/AIDS. This article is not a declaration of finite practices of being an ally to these individuals. Rather, it draws from personal reflection, public health research, and student affairs theory to make sense of a personal journey where HIV/AIDS has touched nearly every aspect of my life: family, friends, colleagues, and students. This article asks the reader to examine their personal connections, experiences, perceptions and biases of students living with HIV/AIDS, particularly those who are newly diagnosed, in order to be better prepared and informed friends, colleagues, and student affairs practitioners to those living with the disease

Cliques, colouring and satisfiability : from structure to algorithms

We examine the implications of various structural restrictions on the computational complexity of three central problems of theoretical computer science (colourability, independent set and satisfiability), and their relatives. All problems we study are generally NP-hard and they remain NP-hard under various restrictions. Finding the greatest possible restrictions under which a problem is computationally difficult is important for a number of reasons. Firstly, this can make it easier to establish the NP-hardness of new problems by allowing easier transformations. Secondly, this can help clarify the boundary between tractable and intractable instances of the problem. Typically an NP-hard graph problem admits an infinite sequence of narrowing families of graphs for which the problem remains NP-hard. We obtain a number of such results; each of these implies necessary conditions for polynomial-time solvability of the respective problem in restricted graph classes. We also identify a number of classes for which these conditions are sufficient and describe explicit algorithms that solve the problem in polynomial time in those classes. For the satisfiability problem we use the language of graph theory to discover the very first boundary property, i.e. a property that separates tractable and intractable instances of the problem. Whether this property is unique remains a big open problem

Subgraph complementation and minimum rank

Any finite simple graph $G = (V,E)$ can be represented by a collection $\mathscr{C}$ of subsets of $V$ such that $uv\in E$ if and only if $u$ and $v$ appear together in an odd number of sets in $\mathscr{C}$. Let $c_2(G)$ denote the minimum cardinality of such a collection. This invariant is equivalent to the minimum dimension of a faithful orthogonal representation of $G$ over $\mathbb{F}_2$ and is closely connected to the minimum rank of $G$. We show that $c_2(G) = \operatorname{mr}(G,\mathbb{F}_2)$ when $\operatorname{mr}(G,\mathbb{F}_2)$ is odd, or when $G$ is a forest. Otherwise, $\operatorname{mr}(G,\mathbb{F}_2)\leq c_2(G)\leq \operatorname{mr}(G,\mathbb{F}_2)+1$. Furthermore, we show that the following are equivalent for any graph $G$ with at least one edge: i. $c_2(G)=\operatorname{mr}(G,\mathbb{F}_2)+1$; ii. the adjacency matrix of $G$ is the unique matrix of rank $\operatorname{mr}(G,\mathbb{F}_2)$ which fits $G$ over $\mathbb{F}_2$; iii. there is a minimum collection $\mathscr{C}$ as described in which every vertex appears an even number of times; and iv. for every component $G'$ of $G$, $c_2(G') = \operatorname{mr}(G',\mathbb{F}_2) + 1$. We also show that, for these graphs, $\operatorname{mr}(G,\mathbb{F}_2)$ is twice the minimum number of tricliques whose symmetric difference of edge sets is $E$. Additionally, we provide a set of upper bounds on $c_2(G)$ in terms of the order, size, and vertex cover number of $G$. Finally, we show that the class of graphs with $c_2(G)\leq k$ is hereditary and finitely defined. For odd $k$, the sets of minimal forbidden induced subgraphs are the same as those for the property $\operatorname{mr}(G,\mathbb{F}_2)\leq k$, and we exhibit this set for $c_2(G)\leq2$

VELOCITY MEASUREMENTS IN RESERVOIR ROCK SAMPLES FROM A LIMESTONE UNIT USING VARIOUS PORE FLUIDS, AND INTEGRATION WITH WELL LOGS AND SEISMIC DATA

One of the most promising methods proposed to mitigate excess global CO2 is carbon sequestration, a process in which CO2 is pressurized and injected into geologic formations. A technical challenge surrounding the geologic sequestration of CO2 is tracking the movement of the fluids pumped underground. Monitoring, verification and accounting activities related to CO2 storage are important for assuring that sequestered CO2 does not escape to the surface. Tracking this carbon dioxide can be considerably aided by reflection seismic-based detection methods. This thesis employs lab scale velocity measurements of core samples, under in situ reservoir pressure and temperature conditions, combined with multiple 3D reflection seismic surveys, to effectively track the movements of CO2 after injection. The National Energy Technology Laboratory (NETL) of the United States Department of Energy began to participate in research of an enhanced oil recovery project including the injection of CO2 deep into a reservoir structure, repeat reflection seismic surveys, collection of well logs, and rock physics analysis of sample core material. Our study is concentrated on a small area of this field around the injection site. At this site, hydrocarbons were previously moved via water injection. We obtained ultrasonic elastic wave velocity measurements that were conducted under several different saturation scenarios, including CO2 saturated samples, so a quantification of the conditions in different parts of the reservoir could be determined. This approach can help to characterize what is taking place inside the reservoir. Core-scale velocity measurements under in situ conditions allow us to predict changes in future well log or seismic surveys. The large amounts of CO2 accumulated over the past four decades in this reservoir give us a real world example of how an EOR site matures. Combining core scale, well log scale, and seismic scale measurements allows a better understanding of the various processes at work when CO2 is sequestered in a limestone reservoir

Second-generation PLINK: rising to the challenge of larger and richer datasets

PLINK 1 is a widely used open-source C/C++ toolset for genome-wide association studies (GWAS) and research in population genetics. However, the steady accumulation of data from imputation and whole-genome sequencing studies has exposed a strong need for even faster and more scalable implementations of key functions. In addition, GWAS and population-genetic data now frequently contain probabilistic calls, phase information, and/or multiallelic variants, none of which can be represented by PLINK 1's primary data format. To address these issues, we are developing a second-generation codebase for PLINK. The first major release from this codebase, PLINK 1.9, introduces extensive use of bit-level parallelism, O(sqrt(n))-time/constant-space Hardy-Weinberg equilibrium and Fisher's exact tests, and many other algorithmic improvements. In combination, these changes accelerate most operations by 1-4 orders of magnitude, and allow the program to handle datasets too large to fit in RAM. This will be followed by PLINK 2.0, which will introduce (a) a new data format capable of efficiently representing probabilities, phase, and multiallelic variants, and (b) extensions of many functions to account for the new types of information. The second-generation versions of PLINK will offer dramatic improvements in performance and compatibility. For the first time, users without access to high-end computing resources can perform several essential analyses of the feature-rich and very large genetic datasets coming into use.Comment: 2 figures, 1 additional fil

Evaporation of Sessile Drops under Combined Diffusion and Natural Convection

Experiments were conducted to investigate the range of applicability of a commonly used assumption for evaporation models of sessile drops, that the transport mechanism that controls the evaporation is vapor diffusion. The evaporation rates of sessile drops of 3-methylpentane, hexane, cyclohexane, and heptane were measured. The radius of the drop contact line was constant during the measurements and drops of radius from 1 mm to 22 mm were studied. It was found that a diffusion-controlled evaporation model underpredicts the evaporation rate from 36% to 80% depending on the drop size. The increase in the evaporation rate was attributed to a second transport mechanism, natural convection of the vapors, and an empirical model was developed for conditions of combined diffusive and convective transport. Over the broad range of volatilities and drop sizes studied, the evaporation rates computed using the combined transport model agree with the measured values with less than 6% root mean square error