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 $n$

Magnitude cohomology

Magnitude homology was introduced by Hepworth and Willerton in the case of graphs, and was later extended by Leinster and Shulman to metric spaces and enriched categories. Here we introduce the dual theory, magnitude cohomology, which we equip with the structure of an associative unital graded ring. Our first main result is a 'recovery theorem' showing that the magnitude cohomology ring of a finite metric space completely determines the space itself. The magnitude cohomology ring is non-commutative in general, for example when applied to finite metric spaces, but in some settings it is commutative, for example when applied to ordinary categories. Our second main result explains this situation by proving that the magnitude cohomology ring of an enriched category is graded-commutative whenever the enriching category is cartesian. We end the paper by giving complete computations of magnitude cohomology rings for several large classes of graphs.Comment: 27 page

Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of $n$ vertices and $h$ holes. We introduce a \emph{graph of oriented distances} to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in $\min \{\,O(n^\omega), O(n^2 + nh \log h + \chi^2)\,\}$ time, where $\omega<2.373$ denotes the matrix multiplication exponent and $\chi\in \Omega(n)\cap O(n^2)$ is the number of edges of the graph of oriented distances. We also provide a faster algorithm for computing the diameter that runs in $O(n^2 \log n)$ time

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