Chromatic index of graphs with no cycle with a unique chord

The class C of graphs that do not contain a cycle with a unique chord was recently studied by Trotignon and Vušković (in press) [23], who proved for these graphs strong structure results which led to solving the recognition and vertex-colouring problems in polynomial time. In the present paper, we investigate how these structure results can be applied to solve the edge-colouring problem in the class. We give computational complexity results for the edge-colouring problem restricted to C and to the subclass C′ composed of the graphs of C that do not have a 4-hole. We show that it is NP-complete to determine whether the chromatic index of a graph is equal to its maximum degree when the input is restricted to regular graphs of C with fixed degree Δ≥3. For the subclass C′, we establish a dichotomy: if the maximum degree is Δ=3, the edge-colouring problem is NP-complete, whereas, if Δ≠3, the only graphs for which the chromatic index exceeds the maximum degree are the odd holes and the odd order complete graphs, a characterization that solves edge-colouring problem in polynomial time. We determine two subclasses of graphs in C′ of maximum degree 3 for which edge-colouring is polynomial. Finally, we remark that a consequence of one of our proofs is that edge-colouring in NP-complete for r-regular tripartite graphs of degree Δ≥3, for r≥3

Jensen–Shannon Distance-Based Filter and Unsupervised Evaluation Metrics for Polarimetric Weather Radar Processing

An effective filtering technique is required for rainfall rate measurement by weather radar. A Jensen–Shannon distance (JSD)-based thresholding filter is proposed to mitigate nonmeteorological signals, either in clear air or rain situations. This algorithm classifies range-Doppler bins into two classes, hydrometeors and nonhydrometeors, based on spectral polarimetric variable features. The result is a mask to be applied on the spectrograms. The variable selected here is the spectral co-polar correlation coefficient, available in dual-polarization and full polarimetric radars. The algorithm first does global thresholding by finding an optimized threshold value based on the averaged clear-air spectral polarimetric variable distribution. Next, classical filtering steps are carried out like a ground clutter notch filter around 0 ms−1, a mathematical morphology to fill gaps in the hydrometeor areas, and a removal of narrow Doppler power spectra. The second part of this article is the assessment of filtering techniques without ground truth. An assessment without ground truth is useful to select optimal algorithm configurations from a large solution space. Criteria of good filtering are defined both in the spectral and time domain. Based on those criteria, subjective and objective unsupervised evaluation metrics are derived, with a focus on the objective ones. Data, including clear air and rain collected from a full polarimetric Doppler X-band radar in the urban area, are used. With the proposed unsupervised evaluation metrics, the JSD-based thresholding filter is compared to two spectral polarimetric filters. Overall, the JSD-based filter performs very well considering both the subjective and the objective evaluation metrics.Accepted author manuscriptAtmospheric Remote SensingMicrowave Sensing, Signals & System

Evolution of X-ray galaxy Cluster Properties in a Representative Sample (EXCPReS). Optimal binning for temperature profile extraction

International audienceWe present XMM-Newton observations of a representative X-ray selected sample of 31 galaxy clusters at moderate redshift $(0.4<z<0.6)$, spanning the mass range $10^{14} < M_{\textrm 500} < 10^{15}$~M$_\odot$. This sample, EXCPRES (Evolution of X-ray galaxy Cluster Properties in a Representative Sample), is used to test and validate a new method to produce optimally-binned cluster X-ray temperature profiles. The method uses a dynamic programming algorithm, based on partitioning of the soft-band X-ray surface brightness profile, to obtain a binning scheme that optimally fulfills a given signal-to-noise threshold criterion out to large radius. From the resulting optimally-binned EXCPRES temperature profiles, and combining with those from the local REXCESS sample, we provide a generic scaling relation between the relative error on the temperature and the [0.3-2] keV surface brightness signal-to-noise ratio, and its dependence on temperature and redshift. We derive an average scaled 3D temperature profile for the sample. Comparing to the average scaled 3D temperature profiles from REXCESS, we find no evidence for evolution of the average profile shape within the redshift range that we probe

Evolution of X-ray galaxy Cluster Properties in a Representative Sample (EXCPReS). Optimal binning for temperature profile extraction

International audienceWe present XMM-Newton observations of a representative X-ray selected sample of 31 galaxy clusters at moderate redshift $(0.4<z<0.6)$, spanning the mass range $10^{14} < M_{\textrm 500} < 10^{15}$~M$_\odot$. This sample, EXCPRES (Evolution of X-ray galaxy Cluster Properties in a Representative Sample), is used to test and validate a new method to produce optimally-binned cluster X-ray temperature profiles. The method uses a dynamic programming algorithm, based on partitioning of the soft-band X-ray surface brightness profile, to obtain a binning scheme that optimally fulfills a given signal-to-noise threshold criterion out to large radius. From the resulting optimally-binned EXCPRES temperature profiles, and combining with those from the local REXCESS sample, we provide a generic scaling relation between the relative error on the temperature and the [0.3-2] keV surface brightness signal-to-noise ratio, and its dependence on temperature and redshift. We derive an average scaled 3D temperature profile for the sample. Comparing to the average scaled 3D temperature profiles from REXCESS, we find no evidence for evolution of the average profile shape within the redshift range that we probe