On the Wiener Index of Orientations of Graphs

The Wiener index of a strong digraph $D$ is defined as the sum of the distances between all ordered pairs of vertices. This definition has been extended to digraphs that are not necessarily strong by defining the distance from a vertex $a$ to a vertex $b$ as $0$ if there is no path from $a$ to $b$ in $D$. Knor, \u{S}krekovski and Tepeh [Some remarks on Wiener index of oriented graphs. Appl.\ Math.\ Comput.\ {\bf 273}] considered orientations of graphs with maximum Wiener index. The authors conjectured that for a given tree $T$, an orientation $D$ of $T$ of maximum Wiener index always contains a vertex $v$ such that for every vertex $u$, there is either a $(u,v)$-path or a $(v,u)$-path in $D$. In this paper we disprove the conjecture. We also show that the problem of finding an orientation of maximum Wiener index of a given graph is NP-complete, thus answering a question by Knor, \u{S}krekovski and Tepeh [Orientations of graphs with maximum Wiener index. Discrete Appl.\ Math.\ 211]. We briefly discuss the corresponding problem of finding an orientation of minimum Wiener index of a given graph, and show that the special case of deciding if a given graph on $m$ edges has an orientation of Wiener index $m$ can be solved in time quadratic in $n$

Fast and Compact Exact Distance Oracle for Planar Graphs

For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an $O(n^{5/3})$-space distance oracle which answers exact distance queries in $O(\log n)$ time for $n$-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space i.e., space $O(n^{2 - \epsilon})$ for some constant $\epsilon > 0$) either required query time polynomial in $n$ or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any $S \ge n^{3/2}$, we show how to obtain an $S$-space distance oracle that answers queries in time $O((n^{5/2}/ S^{3/2}) \log n)$. This is a polynomial improvement over the previous planar distance oracles with $o(n^{1/4})$ query time

Maximized Posteriori Attributes Selection from Facial Salient Landmarks for Face Recognition

This paper presents a robust and dynamic face recognition technique based on the extraction and matching of devised probabilistic graphs drawn on SIFT features related to independent face areas. The face matching strategy is based on matching individual salient facial graph characterized by SIFT features as connected to facial landmarks such as the eyes and the mouth. In order to reduce the face matching errors, the Dempster-Shafer decision theory is applied to fuse the individual matching scores obtained from each pair of salient facial features. The proposed algorithm is evaluated with the ORL and the IITK face databases. The experimental results demonstrate the effectiveness and potential of the proposed face recognition technique also in case of partially occluded faces.Comment: 8 pages, 2 figure

Unsupervised cryo-EM data clustering through adaptively constrained K-means algorithm

In single-particle cryo-electron microscopy (cryo-EM), K-means clustering algorithm is widely used in unsupervised 2D classification of projection images of biological macromolecules. 3D ab initio reconstruction requires accurate unsupervised classification in order to separate molecular projections of distinct orientations. Due to background noise in single-particle images and uncertainty of molecular orientations, traditional K-means clustering algorithm may classify images into wrong classes and produce classes with a large variation in membership. Overcoming these limitations requires further development on clustering algorithms for cryo-EM data analysis. We propose a novel unsupervised data clustering method building upon the traditional K-means algorithm. By introducing an adaptive constraint term in the objective function, our algorithm not only avoids a large variation in class sizes but also produces more accurate data clustering. Applications of this approach to both simulated and experimental cryo-EM data demonstrate that our algorithm is a significantly improved alterative to the traditional K-means algorithm in single-particle cryo-EM analysis.Comment: 35 pages, 14 figure

Revisiting the Impact of Atmospheric Dispersion and Differential Refraction on Widefield Multiobject Spectroscopic Observations. From VLT/VIMOS to Next Generation Instruments

(Abridged) Atmospheric dispersion and field differential refraction impose severe constraints on widefield MOS observations. Flux reduction and spectral distortions must be minimised by a careful planning of the observations -- which is especially true for instruments that use slits instead of fibres. This is the case of VIMOS at the VLT, where MOS observations have been restricted, since the start of operations, to a narrow two-hour range from the meridian to minimise slit losses. We revisit in detail the impact of atmospheric effects on the quality of VIMOS-MOS spectra. We model slit losses across the entire VIMOS FOV as a function of target declination. We explore two different slit orientations at the meridian: along the parallactic angle (North-South), and perpendicular to it (East-West). We show that, for fields culminating at zenith distances larger than 20 deg, slit losses are minimised with slits oriented along the parallactic angle at the meridian. The two-hour angle rule holds for these observations using N-S orientations. Conversely, for fields with zenith angles smaller than 20 deg at culmination, losses are minimised with slits oriented perpendicular to the parallactic angle at the meridian. MOS observations can be effectively extended to plus/minus three hours from the meridian in these cases. In general, night-long observations of a single field will benefit from using the E-W orientation. All-sky or service mode observations, however, require a more elaborate planning that depends on the target declination, and the hour angle of the observations. We establish general rules for the alignment of slits in MOS observations that will increase target observability, enhance the efficiency of operations, and speed up the completion of programmes -- a particularly relevant aspect for the forthcoming spectroscopic public surveys with VIMOS.Comment: Accepted to A&A. 11 pages, 15 figures. This paper presents the new recommendations for optimal slit alignment in VLT/VIMOS observation

Switching Reconstruction of Digraphs

Switching about a vertex in a digraph means to reverse the direction of every edge incident with that vertex. Bondy and Mercier introduced the problem of whether a digraph can be reconstructed up to isomorphism from the multiset of isomorphism types of digraphs obtained by switching about each vertex. Since the largest known non-reconstructible oriented graphs have 8 vertices, it is natural to ask whether there are any larger non-reconstructible graphs. In this paper we continue the investigation of this question. We find that there are exactly 44 non-reconstructible oriented graphs whose underlying undirected graphs have maximum degree at most 2. We also determine the full set of switching-stable oriented graphs, which are those graphs for which all switchings return a digraph isomorphic to the original