On retracts, absolute retracts, and folds in cographs

Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the problem becomes tractable in polynomial time. The problem is also soluble when one cograph is given as an induced subgraph of the other. We characterize absolute retracts of cographs.Comment: 15 page

Snapshot hyperspectral imaging : near-infrared image replicating imaging spectrometer and achromatisation of Wollaston prisms

Conventional hyperspectral imaging (HSI) techniques are time-sequential and rely on temporal scanning to capture hyperspectral images. This temporal constraint can limit the application of HSI to static scenes and platforms, where transient and dynamic events are not expected during data capture. The Near-Infrared Image Replicating Imaging Spectrometer (N-IRIS) sensor described in this thesis enables snapshot HSI in the short-wave infrared (SWIR), without the requirement for scanning and operates without rejection in polarised light. It operates in eight wavebands from 1.1μm to 1.7μm with a 2.0° diagonal field-of-view. N-IRIS produces spectral images directly, without the need for prior topographic or image reconstruction. Additional benefits include compactness, robustness, static operation, lower processing overheads, higher signal-to-noise ratio and higher optical throughput with respect to other HSI snapshot sensors generally. This thesis covers the IRIS design process from theoretical concepts to quantitative modelling, culminating in the N-IRIS prototype designed for SWIR imaging. This effort formed the logical step in advancing from peer efforts, which focussed upon the visible wavelengths. After acceptance testing to verify optical parameters, empirical laboratory trials were carried out. This testing focussed on discriminating between common materials within a controlled environment as proof-of-concept. Significance tests were used to provide an initial test of N-IRIS capability in distinguishing materials with respect to using a conventional SWIR broadband sensor. Motivated by the design and assembly of a cost-effective visible IRIS, an innovative solution was developed for the problem of chromatic variation in the splitting angle (CVSA) of Wollaston prisms. CVSA introduces spectral blurring of images. Analytical theory is presented and is illustrated with an example N-IRIS application where a sixfold reduction in dispersion is achieved for wavelengths in the region 400nm to 1.7μm, although the principle is applicable from ultraviolet to thermal-IR wavelengths. Experimental proof of concept is demonstrated and the spectral smearing of an achromatised N-IRIS is shown to be reduced by an order of magnitude. These achromatised prisms can provide benefits to areas beyond hyperspectral imaging, such as microscopy, laser pulse control and spectrometry

A hyperspectral imaging system for mapping haemoglobin and cytochrome-c-oxidase concentration changes in the exposed cerebral cortex

We present a novel hyperspectral imaging (HSI) system using visible and near-infrared (NIR) light on the exposed cerebral cortex of animals, to monitor and quantify in vivo changes in the oxygenation of haemoglobin and in cellular metabolism via measurement of the redox states of cytochrome-c-oxidase (CCO). The system, named hNIR, is based on spectral scanning illumination at 11 bands (600, 630, 665, 784, 800, 818, 835, 851, 868, 881 and 894 nm), using a supercontinuum laser coupled with a rotating Pellin-Broca prism. Image reconstruction is performed with the aid of a Monte Carlo framework for photon pathlength estimation and post-processing correction of partial volume effects. The system is validated on liquid optical phantoms mimicking brain tissue haemodynamics and metabolism, and finally applied in vivo on the exposed cortex of mice undergoing alternating oxygenation challenges. The results of the study demonstrate the capacity of hNIR to map and quantify the haemodynamic and metabolic states of the exposed cortex at microvascular levels. This represents (to the best of our knowledge) the first example of simultaneous mapping and quantification of cerebral haemoglobin and CCO in vivo using visible and NIR HSI, which can potentially become a powerful tool for better understanding brain physiology

Quorum Colorings of Graphs

Let $G = (V,E)$ be a graph. A partition $\pi = \{V_1, V_2, \ldots, V_k \}$ of the vertices $V$ of $G$ into $k$ {\it color classes} $V_i$, with $1 \leq i \leq k$, is called a {\it quorum coloring} if for every vertex $v \in V$, at least half of the vertices in the closed neighborhood $N[v]$ of $v$ have the same color as $v$. In this paper we introduce the study of quorum colorings of graphs and show that they are closely related to the concept of defensive alliances in graphs. Moreover, we determine the maximum quorum coloring of a hypercube

Hyperspectral imaging of the haemodynamic and metabolic states of the exposed cortex

A hyperspectral imaging (HSI) system to measure and quantify in vivo haemodynamic and metabolic signals from the exposed cerebral cortex of small animals was designed, developed and investigated in this thesis. Imaging brain tissue at multiple narrow wavelength bands in the visible and near-infrared (NIR) range allows one not only to monitor cerebral oxygenation and haemodynamics via mapping of haemoglobin concentration changes, but also to directly target the spatial quantification of cerebral metabolic activity via measurement of the redox states of mitochondrial cytochrome-c-oxidase (CCO). Having both these sets of information in vivo at high resolution on the exposed cortex can provide impactful insight on brain physiology and can help validate corresponding data acquired non-invasively using broadband near-infrared spectroscopy (bNIRS). Several designs and HSI configurations were assessed and compared, including different customised benchtop setups. In the end, a bespoke spectral-scanning HSI system called hNIR, using a supercontinuum laser coupled with a rotating Pellin-Broca prism and a scientific complementary metal-oxide semiconductor (sCMOS) camera, was built, characterised and validated on liquid optical phantoms. In addition, an in-house Monte Carlo (MC) framework for simulating HSI of the haemodynamic and metabolic states of the exposed cortex was also developed using an open-source MC code package (Mesh-based Monte Carlo) and integrated with hNIR, for aiding image reconstruction and enhance quantification, as well as to run computational investigations on the performances of HSI for brain haemodynamic and metabolic monitoring. hNIR was finally applied in vivo on the exposed cerebral cortex of three mice during different levels of hyperoxic and hypoxic stimulation, demonstrating its capability to retrieve high resolution and accurate maps of the relative changes in the concentrations of oxyhaemoglobin (HbO₂), deoxyhaemoglobin (HHb) and the oxidative state of CCO (oxCCO)

On the oriented chromatic number of dense graphs

Let $G$ be a graph with $n$ vertices, $m$ edges, average degree $\delta$, and maximum degree $\Delta$. The \emph{oriented chromatic number} of $G$ is the maximum, taken over all orientations of $G$, of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which $\delta\geq\log n$. We prove that every such graph has oriented chromatic number at least $\Omega(\sqrt{n})$. In the case that $\delta\geq(2+\epsilon)\log n$, this lower bound is improved to $\Omega(\sqrt{m})$. Through a simple connection with harmonious colourings, we prove a general upper bound of \Oh{\Delta\sqrt{n}} on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when $G$ is ($c\log n$)-regular for some constant $c>2$, in which case the oriented chromatic number is between $\Omega(\sqrt{n\log n})$ and $\mathcal{O}(\sqrt{n}\log n)$

Forbidden subgraphs and complete partitions

A graph is called an $(r,k)$-graph if its vertex set can be partitioned into $r$ parts of size at most $k$ with at least one edge between any two parts. Let $f(r,H)$ be the minimum $k$ for which there exists an $H$-free $(r,k)$-graph. In this paper we build on the work of Axenovich and Martin, obtaining improved bounds on this function when $H$ is a complete bipartite graph, even cycle, or tree. Some of these bounds are best possible up to a constant factor and confirm a conjecture of Axenovich and Martin in several cases. We also generalize this extremal problem to uniform hypergraphs and prove some initial results in that setting

Hyperspectral Imaging of the Hemodynamic and Metabolic States of the Exposed Cortex: Investigating a Commercial Snapshot Solution

Hyperspectral imaging (HSI) systems have the potential to retrieve in vivo hemodynamic and metabolic signals from the exposed cerebral cortex. The use of multiple narrow wavelength bands in the near infrared (NIR) range theoretically allows not only to image brain tissue oxygenation and hemodynamics via mapping of hemoglobin concentration changes, but also to directly quantify cerebral metabolism via measurement of the redox states of mitochondrial cytochrome-c-oxidase (CCO). The aim of this study is to assess the possibility of performing hyperspectral imaging of in vivo cerebral oxyhemoglobin (HbO2), deoxyhemoglobin (HHb) and oxidized CCO (oxCCO) using commercially available HSI devices. For this reason, a hyperspectral snapshot solution based on Cubert GmbH technology (S185 FireflEYE camera) has been tested on the exposed cortex of mice during normoxic, hypoxic and hyperoxic conditions. The system allows simultaneous acquisition of 138 wavelength bands between 450 and 998 nm, with spectral sampling and resolution of ~4 to 8 nm. From the hyperspectral data, relative changes in concentration of hemoglobin and oxCCO are estimated and hemodynamic and metabolic maps of the imaged cortex are calculated for two different NIR spectral ranges. Spectroscopic analysis at particular regions of interest is also performed, showing typical oxygen-dependent hemodynamic responses. The results highlight some of the potentials of the technology, but also the limitations of the tested commercial solution for such specific application, in particular regarding spatial resolution