A note on "Folding wheels and fans."

In S.Gervacio, R.Guerrero and H.Rara, Folding wheels and fans, Graphs and Combinatorics 18 (2002) 731-737, the authors obtain formulas for the clique numbers onto which wheels and fans fold. We present an interpolation theorem which generalizes their theorems 4.2 and 5.2. We show that their formula for wheels is wrong. We show that for threshold graphs, the achromatic number and folding number coincides with the chromatic number

Complete Acyclic Colorings

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure

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

Achromatizing a liquid-crystal spectropolarimeter: Retardance vs Stokes-based calibration of HiVIS

Astronomical spectropolarimeters can be subject to many sources of systematic error which limit the precision and accuracy of the instrument. We present a calibration method for observing high-resolution polarized spectra using chromatic liquid-crystal variable retarders (LCVRs). These LCVRs allow for polarimetric modulation of the incident light without any moving optics at frequencies >10Hz. We demonstrate a calibration method using pure Stokes input states that enables an achromatization of the system. This Stokes-based deprojection method reproduces input polarization even though highly chromatic instrument effects exist. This process is first demonstrated in a laboratory spectropolarimeter where we characterize the LCVRs and show example deprojections. The process is then implemented the a newly upgraded HiVIS spectropolarimeter on the 3.67m AEOS telescope. The HiVIS spectropolarimeter has also been expanded to include broad-band full-Stokes spectropolarimetry using achromatic wave-plates in addition to the tunable full-Stokes polarimetric mode using LCVRs. These two new polarimetric modes in combination with a new polarimetric calibration unit provide a much more sensitive polarimetric package with greatly reduced systematic error.Comment: Accepted in PAS

A Neuromorphic Model for Achromatic and Chromatic Surface Representation of Natural Images

This study develops a neuromorphic model of human lightness perception that is inspired by how the mammalian visual system is designed for this function. It is known that biological visual representations can adapt to a billion-fold change in luminance. How such a system determines absolute lightness under varying illumination conditions to generate a consistent interpretation of surface lightness remains an unsolved problem. Such a process, called "anchoring" of lightness, has properties including articulation, insulation, configuration, and area effects. The model quantitatively simulates such psychophysical lightness data, as well as other data such as discounting the illuminant, the double brilliant illusion, and lightness constancy and contrast effects. The model retina embodies gain control at retinal photoreceptors, and spatial contrast adaptation at the negative feedback circuit between mechanisms that model the inner segment of photoreceptors and interacting horizontal cells. The model can thereby adjust its sensitivity to input intensities ranging from dim moonlight to dazzling sunlight. A new anchoring mechanism, called the Blurred-Highest-Luminance-As-White (BHLAW) rule, helps simulate how surface lightness becomes sensitive to the spatial scale of objects in a scene. The model is also able to process natural color images under variable lighting conditions, and is compared with the popular RETINEX model.Air Force Office of Scientific Research (F496201-01-1-0397); Defense Advanced Research Project and the Office of Naval Research (N00014-95-0409, N00014-01-1-0624

Some colouring problems for Paley graphs

The Paley graph Pq, where q≡1(mod4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x-y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion. © 2005 Elsevier B.V. All rights reserved