Kernels by Monochromatic Directed Paths in 3-Colored Tournaments and Quasi-Tournaments
We call the tournament T an m-coloured tournament if the arcs of T are coloured with m colours. If v is a vertex of an m-coloured tournament T, we denote by ζ(v) the set of colours assigned to the arcs with v as endpoint, ζ − (v) the set of colours assigned to the arcs with v as final vertex, and ζ + (v) the set of colours assigned to the arcs with v as an initial vertex. In this paper, we prove that if T is a 3-coloured tournament which does not contain C3 such that: 1. for each v ∈ V(T) |ζ − (v) |≤ 2,
Recent problems and results about kernels in directed graphs
Abstract In Section 1, we survey the existence theorems for a kernel; in Section 2, we discuss a new conjecture which could constitute a bridge between the kernel problems and the perfect graph conjecture. In fact, we believe that a graph is ‘quasi-perfect’ if and only if it is perfect.
Multivariate Image Texture by Multivariate Variogram for Multispectral Image Classification
Traditional image texture measure usually allows a texture description of a single band of the spectrum, characterizing the spatial variability of gray-level values within the singleband image. A problem with the approach while applied to multispectral images is that it only uses the texture information from selected bands. In this paper, we propose a new multivariate texture measure based on the multivariate variogram. The multivariate texture is computed from all bands of a multispectral image, which characterizes the multivariate spatial autocorrelation among those bands. In order to evaluate the performance of the proposed texture measure, the derived multivariate texture image is combined with the spectral data in image classification. The result is compared to classifications using spectral data alone and plus traditional texture images. A machine learning classifier based on Support Vector Machines (SVMs) is used for image classification. The experimental results demonstrate that the inclusion of multivariate texture information in multispectral image classification significantly improves the overall accuracy, with 5 to 13.5 percent of improvement, compared to the classification with spectral information alone. The results also show that when incorporated in image classification as an additional band, the multivariate texture results in high overall accuracy, which is comparable with or higher than the best results from the existing single-band and two-band texture measures, such as the variogram, cross variogram and Gray-Level Co-occurrence Matrix (GLCM) based texture. Overall, the multivariate texture provides the useful spatial information for land-cover classification, which is different from the traditional single band texture. Moreover, it avoids the band selection procedure which is prerequisite to traditional texture computation and would help to achieve high accuracy in the most classification tasks.