Colorful Strips

Given a planar point set and an integer $k$, we wish to color the points with $k$ colors so that any axis-aligned strip containing enough points contains all colors. The goal is to bound the necessary size of such a strip, as a function of $k$. We show that if the strip size is at least $2k{-}1$, such a coloring can always be found. We prove that the size of the strip is also bounded in any fixed number of dimensions. In contrast to the planar case, we show that deciding whether a 3D point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. We also consider the problem of coloring a given set of axis-aligned strips, so that any sufficiently covered point in the plane is covered by $k$ colors. We show that in $d$ dimensions the required coverage is at most $d(k{-}1)+1$. Lower bounds are given for the two problems. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. Finally, we study a variant where strips are replaced by wedges

What\u27s going on, edition 79-6 (October 29, 1979)

https://egrove.olemiss.edu/aicpa_news/2355/thumbnail.jp

The B.G. News April 11, 1958

The BGSU campus student newspaper April 11, 1958. Volume 42 - Issue 38https://scholarworks.bgsu.edu/bg-news/2412/thumbnail.jp

Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Color categorization in infants

In human infants trichromatic vision is functional within the first few months of life. Infants also make categorical responses to color – appearing to group together similar colors, but with distinct boundaries. Recent developments have revealed a candidate neural basis for infant color categories – the low-level cone-opponent mechanisms of color vision. These pre-linguistic boundaries appear to drive infant looking behavior, and may provide discontinuity in color perception around which linguistic color categories are formed in adults. This finding opens up new avenues for research, such as the need to understand how color categorization develops from being based on the low-level mechanisms of color vision in infancy to reflecting the linguistic, cultural and visual environment of the individual

Red, yellow, green and blue are not particularly colorful

Colorfulness and saturation have been neglected in research on color appearance and color naming. Perceptual particularities, such as cross-cultural stability, “focality”, “uniqueness”, “salience” and “prominence” have been observed for red, yellow, green, and blue, when those colors were more saturated than other colors in the stimulus samples. The present study tests whether high saturation is a particular property of red, yellow, green and blue, which would explain those observations. First, we carefully determined the category prototypes and unique hues for red, yellow, green, and blue. Using different approaches in two experiments, we assessed discriminable saturation as the number of just-noticeable differences away from the adaptation point (i.e. neutral gray). Results show that some hues can reach much higher levels of maximal saturation than others. However, typical and unique red, yellow, green, and blue are not particularly colorful. Many other, intermediate colors have a larger range of discriminable saturation than these colors. These findings suggest that prior claims of perceptual salience of category prototypes and unique hues actually reflect biases in stimulus sets rather than perceptual properties. Additional analyses show that consistent prototype choices across fundamentally different languages are strongly related to the variation of discriminable saturation in the stimulus sets. Our findings also undermine the idea that every color can be produced by a mixture of unique hues. Finally, the measurements in this study provide a large amount of data on saturation across hues, which allows for reevaluating existing estimates of saturation in future studies

Crossing-Optimal Extension of Simple Drawings

In extension problems of partial graph drawings one is given an incomplete drawing of an input graph G and is asked to complete the drawing while maintaining certain properties. A prominent area where such problems arise is that of crossing minimization. For plane drawings and various relaxations of these, there is a number of tractability as well as lower-bound results exploring the computational complexity of crossing-sensitive drawing extension problems. In contrast, comparatively few results are known on extension problems for the fundamental and broad class of simple drawings, that is, drawings in which each pair of edges intersects in at most one point. In fact, the extension problem of simple drawings has only recently been shown to be NP-hard even for inserting a single edge. In this paper we present tractability results for the crossing-sensitive extension problem of simple drawings. In particular, we show that the problem of inserting edges into a simple drawing is fixed-parameter tractable when parameterized by the number of edges to insert and an upper bound on newly created crossings. Using the same proof techniques, we are also able to answer several closely related variants of this problem, among others the extension problem for k-plane drawings. Moreover, using a different approach, we provide a single-exponential fixed-parameter algorithm for the case in which we are only trying to insert a single edge into the drawing