Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets

An identifying code of a (di)graph $G$ is a dominating subset $C$ of the vertices of $G$ such that all distinct vertices of $G$ have distinct (in)neighbourhoods within $C$. In this paper, we classify all finite digraphs which only admit their whole vertex set in any identifying code. We also classify all such infinite oriented graphs. Furthermore, by relating this concept to a well known theorem of A. Bondy on set systems we classify the extremal cases for this theorem

Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) Open Locating-Dominating Set and Metric Dimension) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter 2 and permutation graphs of diameter 2. While Identifying Code and (Open) Locating-Dominating Set are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting Metric Dimension is W[2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable

Sequential Metric Dimension

International audienceSeager introduced the following game in 2013. An invisible and immobile target is hidden at some vertex of a graph $G$. Every step, one vertex $v$ of $G$ can be probed which results in the knowledge of the distance between $v$ and the target. The objective of the game is to minimize the number of steps needed to locate the target, wherever it is. We address the generalization of this game where $k ≥ 1$ vertices can be probed at every step. Our game also generalizes the notion of the metric dimension of a graph. Precisely, given a graph $G$ and two integers $k, ≥ 1$, the Localization Problem asks whether there exists a strategy to locate a target hidden in $G$ in at most steps by probing at most $k$ vertices per step. We show this problem is NP-complete when $k$ (resp.,) is a fixed parameter. Our main results are for the class of trees where we prove this problem is NP-complete when $k$ and are part of the input but, despite this, we design a polynomial-time (+1)-approximation algorithm in trees which gives a solution using at most one more step than the optimal one. It follows that the Localization Problem is polynomial-time solvable in trees if $k$ is fixed

Algorithms and complexity for path covers of temporal DAGs

A path cover of a digraph is a collection of paths collectively containing its vertex set. A path cover with minimum cardinality for a directed acyclic graph can be found in polynomial time [Fulkerson, AMS’56; Cáceres et al., SODA’22]. Moreover, Dilworth’s celebrated theorem on chain coverings of partially ordered sets equivalently states that the minimum size of a path cover of a DAG is equal to the maximum size of a set of mutually unreachable vertices. In this paper, we examine how far these classic results can be extended to a dynamic setting. A temporal digraph has an arc set that changes over discrete time-steps; if the underlying digraph is acyclic, then it is a temporal DAG. A temporal path is a directed path in the underlying digraph, such that the time-steps of arcs are strictly increasing along the path. Two temporal paths are temporally disjoint if they do not occupy any vertex at the same time. A temporal path cover is a collection C of temporal paths that covers all vertices, and C is temporally disjoint if all its temporal paths are pairwise temporally disjoint. We study the computational complexities of the problems of finding a minimum-size temporal (disjoint) path cover (denoted as Temporal Path Cover and Temporally Disjoint Path Cover). On the negative side, we show that both Temporal Path Cover and Temporally Disjoint Path Cover are NP-hard even when the underlying DAG is planar, bipartite, subcubic, and there are only two arc-disjoint time-steps. Moreover, Temporally Disjoint Path Cover remains NP-hard even on temporal oriented trees. We also observe that natural temporal analogues of Dilworth’s theorem on these classes of temporal DAGs do not hold. In contrast, we show that Temporal Path Cover is polynomial-time solvable on temporal oriented trees by a reduction to Clique Cover for (static undirected) weakly chordal graphs (a subclass of perfect graphs for which Clique Cover admits an efficient algorithm). This highlights an interesting algorithmic difference between the two problems. Although it is NP-hard on temporal oriented trees, Temporally Disjoint Path Cover becomes polynomial-time solvable on temporal oriented lines and temporal rooted directed trees. Motivated by the hardness result on trees, we show that, in contrast, Temporal Path Cover admits an XP time algorithm with respect to parameter tmax + tw, where tmax is the maximum time-step and tw is the treewidth of the underlying static undirected graph; moreover, Temporally Disjoint Path Cover admits an FPT algorithm with respect to the same parameterization

Cosmic Shear Statistics and Cosmology

We report a measurement of cosmic shear correlations using an effective area of 6.5 sq. deg. of the VIRMOS deep imaging survey in progress at the Canada-France-Hawaii Telescope. We measured various shear correlation functions, the aperture mass statistic and the top-hat smoothed variance of the shear with a detection significance exceeding 12 sigma for each of them. We present results on angular scales from 3 arc-seconds to half a degree. The consistency of different statistical measures is demonstrated and confirms the lensing origin of the signal through tests that rely on the scalar nature of the gravitational potential. For Cold Dark Matter models we find $\sigma_8 \Omega_0^{0.6}=0.43^{+0.04}_{-0.05}$ at the 95% confidence level. The measurement over almost three decades of scale allows to discuss the effect of the shape of the power spectrum on the cosmological parameter estimation. The degeneracy on sigma_8-Omega_0 can be broken if priors on the shape of the linear power spectrum (that can be parameterized by Gamma) are assumed. For instance, with Gamma=0.21 and at the 95% confidence level, we obtain 0.60.65 and Omega_0<0.4 for flat (Lambda-CDM) models. From the tangential/radial modes decomposition we can set an upper limit on the intrinsic shape alignment, which was recently suggested as a possible contribution to the lensing signal. Within the error bars, there is no detection of intrinsic shape alignment for scales larger than 1'.Comment: 13 pages, submitted to A&

AEGIS: The Diversity of Bright Near-IR Selected Distant Red Galaxies

We use deep and wide near infrared (NIR) imaging from the Palomar telescope combined with DEEP2 spectroscopy and Hubble Space Telescope (HST) and Chandra Space Telescope imaging to investigate the nature of galaxies that are red in NIR colors. We locate these `distant red galaxies' (DRGs) through the color cut (J-K)_{vega} > 2.3 over 0.7 deg^{2}, where we find 1010 DRG candidates down to K_s = 20.5. We combine 95 high quality spectroscopic redshifts with photometric redshifts from BRIJK photometry to determine the redshift and stellar mass distributions for these systems, and morphological/structural and X-ray properties for 107 DRGs in the Extended Groth Strip. We find that many bright (J-K)_{vega}>2.3 galaxies with K_s2 systems massive with M_*>10^{11} M_solar. HST imaging shows that the structural properties and morphologies of DRGs are also diverse, with the majority elliptical/compact (57%), and the remainder edge-on spirals (7%), and peculiar galaxies (29%). The DRGs at z < 1.4 with high quality spectroscopic redshifts are generally compact, with small half light radii, and span a range in rest-frame optical properties. The spectral energy distributions for these objects differ from higher redshift DRGs: they are bluer by one magnitude in observed (I-J) color. A pure IR color selection of high redshift populations is not sufficient to identify unique populations, and other colors, or spectroscopic redshifts are needed to produce homogeneous samples

The VIMOS VLT Deep Survey - First epoch VVDS-Deep survey: 11564 spectra with 17.5<=IAB<=24, and the redshift distribution over 0< z <=5

This paper presents the ``First Epoch'' sample from the VIMOS VLT Deep Survey (VVDS). The VVDS goals, observations, data reduction with VIPGI, and redshift measurement with KBRED are discussed. Data have been obtained with the VIsible Multi Object Spectrograph (VIMOS) on the ESO-VLT UT3, allowing to observe ~600 slits simultaneously at R~230. A total of 11564 objects have been observed in the VVDS-02h and VVDS-CDFS Deep fields over a total area of 0.61deg^2, selected solely on the basis of apparent magnitude 17.5 <=I_{AB} <=24. The VVDS covers the redshift range 0 < z <= 5. It is successfully going through the ``redshift desert'' 1.5<z<2.2, while the range 2.2<z<2.7 remains of difficult access because of the VVDS wavelength coverage.A total of 9677 galaxies have a redshift measurement, 836 are stars, 90 are AGNs, and a redshift could not be measured for 961 objects. There are 1065 galaxies with a measured redshift z>1.4. The survey reaches a redshift measurement completeness of 78% overall (93% including less reliable objects), with a spatial sampling of the population of galaxies of 25% and ~30% in the VVDS-02h and VVDS-CDFS. The redshift accuracy measured from repeated observations with VIMOS and comparison to other surveys is ~276km/s. From this sample we present for the first time the redshift distribution of a magnitude limited spectroscopic sample down to IAB=24. The redshift distribution has a median of z=0.62, z=0.65, z=0.70, and z=0.76, for magnitude limited samples with IAB<=22.5, 23, 23.5, and 24. A high redshift tail above redshift 2 and up to redshift 5 becomes readily apparent for IAB>23.5, probing the bright star forming population of galaxies. This sample provides an unprecedented dataset to study galaxy evolution over 90% of the life of the universeComment: 30 pages, accepted 22-Feb-05 in A&

A Deep Probe of the Galaxy Stellar Mass Functions at z~1-3 with the GOODS NICMOS Survey

We use a sample of 8298 galaxies observed in the HST GOODS NICMOS Survey (GNS) to construct the galaxy stellar mass function as a function of both redshift and stellar mass up to z=3.5 and down to masses of Mstar=10^8.5 Msun at z~1. We discover that a significant fraction of all massive Mstar>10^11 Msun galaxies are in place up to the highest redshifts we probe, with a decreasing fraction of lower mass galaxies present at all redshifts. This is an example of `galaxy mass downsizing', and is the result of massive galaxies forming before lower mass ones, and not just simply ending their star formation earlier as in traditional downsizing scenarios. We find that the faint end slope is significantly steeper than what is found in previous investigations. We demonstrate that this steeper mass function better matches the stellar mass added due to star formation, thereby alleviating some of the mismatch between these two measures of the evolution of galaxy mass. We furthermore examine the stellar mass function divided into blue/red systems, as well as for star forming and non-star forming galaxies. We find a similar mass downsizing present for both blue/red and star-forming/non-star forming galaxies, and that the low mass galaxies are mostly all blue, and are therefore creating the steep mass functions. We furthermore show that, although there is a downsizing such that high mass galaxies are nearer their z=0 values at high redshift, this turns over at masses Mstar~10^10 Msun, such that the lowest mass galaxies are more common than galaxies at slight higher masses, creating a `dip' in the observed galaxy mass function. We argue that the galaxy assembly process may be driven by different mechanisms at low and high masses, and that the efficiency of the galaxy formation process is lowest at masses Mstar~10^10 Msun at 1<z<3. (Abridged)Comment: 16 pages, 11 figures, MNRAS, accepte

Absence of Evidence Is Not Evidence of Absence: The Color-Density Relation at Fixed Stellar Mass Persists to z ~ 1

We use data drawn from the DEEP2 Galaxy Redshift Survey to investigate the relationship between local galaxy density, stellar mass, and rest-frame galaxy color. At z ~ 0.9, we find that the shape of the stellar mass function at the high-mass (log (M*/Msun) > 10.1) end depends on the local environment, with high-density regions favoring more massive systems. Accounting for this stellar mass-environment relation (i.e., working at fixed stellar mass), we find a significant color-density relation for galaxies with 10.6 < log(M*/Msun) < 11.1 and 0.75 < z < 0.95. This result is shown to be robust to variations in the sample selection and to extend to even lower masses (down to log(M*/Msun) ~ 10.4). We conclude by discussing our results in comparison to recent works in the literature, which report no significant correlation between galaxy properties and environment at fixed stellar mass for the same redshift and stellar mass domain. The non-detection of environmental dependence found in other data sets is largely attributable to their smaller samples size and lower sampling density, as well as systematic effects such as inaccurate redshifts and biased analysis techniques. Ultimately, our results based on DEEP2 data illustrate that the evolutionary state of a galaxy at z ~ 1 is not exclusively determined by the stellar mass of the galaxy. Instead, we show that local environment appears to play a distinct role in the transformation of galaxy properties at z > 1.Comment: 10 pages, 5 Figures; Accepted for publication in MNRA

The VIMOS VLT Deep Survey: The build-up of the colour-density relation

We investigate the redshift and luminosity evolution of the galaxy colour-density relation using the data from the First Epoch VIMOS-VLT Deep Survey (VVDS). The size (6582 galaxies), depth (I_AB<=24) and redshift sampling rate of the survey enable us to reconstruct the 3D galaxy environment on relatively local scales (R=5 Mpc) up to z~1.5. Particular attention has been devoted to calibrate a density reconstruction scheme, which factors out survey selection effects and reproduces in an unbiased way the underlying `real' galaxy environment. While at lower redshift we confirm the existence of a steep colour-density relation, with the fraction of the reddest(/bluest) galaxies of the same luminosity increasing(/decreasing) as a function of density, this trend progressively disappears in the highest redshift bins investigated. The rest frame u*-g' colour-magnitude diagram shows a bimodal pattern in both low and high density environments up to z~1.5. We find that the bimodal distribution is not universal but strongly depends upon environment: at lower redshifts the colour-magnitude diagrams in low and high density regions are significantly different while the progressive weakening of the colour-density relation causes the two bimodal distributions to nearly mirror each other in the highest redshift bin investigated. Both the colour-density and the colour-magnitude-density relations appear to be a transient, cumulative product of genetic and environmental factors operating over at least a period of 9 Gyr. These findings support an evolutionary scenario in which star formation/gas depletion processes are accelerated in more luminous objects and in high density environments: star formation activity is shifting with cosmic time towards lower luminosity (downsizing), and out of high density environments.Comment: 17 pages, 10 figures, figures added, accepted by A&