Complete Acyclic Colorings

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure

Kernels for Below-Upper-Bound Parameterizations of the Hitting Set and Directed Dominating Set Problems

In the {\sc Hitting Set} problem, we are given a collection $\cal F$ of subsets of a ground set $V$ and an integer $p$, and asked whether $V$ has a $p$-element subset that intersects each set in $\cal F$. We consider two parameterizations of {\sc Hitting Set} below tight upper bounds: $p=m-k$ and $p=n-k$. In both cases $k$ is the parameter. We prove that the first parameterization is fixed-parameter tractable, but has no polynomial kernel unless coNP$\subseteq$NP/poly. The second parameterization is W[1]-complete, but the introduction of an additional parameter, the degeneracy of the hypergraph $H=(V,{\cal F})$, makes the problem not only fixed-parameter tractable, but also one with a linear kernel. Here the degeneracy of $H=(V,{\cal F})$ is the minimum integer $d$ such that for each $X\subset V$ the hypergraph with vertex set $V\setminus X$ and edge set containing all edges of $\cal F$ without vertices in $X$, has a vertex of degree at most $d.$ In {\sc Nonblocker} ({\sc Directed Nonblocker}), we are given an undirected graph (a directed graph) $G$ on $n$ vertices and an integer $k$, and asked whether $G$ has a set $X$ of $n-k$ vertices such that for each vertex $y\not\in X$ there is an edge (arc) from a vertex in $X$ to $y$. {\sc Nonblocker} can be viewed as a special case of {\sc Directed Nonblocker} (replace an undirected graph by a symmetric digraph). Dehne et al. (Proc. SOFSEM 2006) proved that {\sc Nonblocker} has a linear-order kernel. We obtain a linear-order kernel for {\sc Directed Nonblocker}

SPARCO : a semi-parametric approach for image reconstruction of chromatic objects

The emergence of optical interferometers with three and more telescopes allows image reconstruction of astronomical objects at the milliarcsecond scale. However, some objects contain components with very different spectral energy distributions (SED; i.e. different temperatures), which produces strong chromatic effects on the interferograms that have to be managed with care by image reconstruction algorithms. For example, the gray approximation for the image reconstruction process results in a degraded image if the total (u, v)-coverage given by the spectral supersynthesis is used. The relative flux contribution of the central object and an extended structure changes with wavelength for different temperatures. For young stellar objects, the known characteristics of the central object (i.e., stellar SED), or even the fit of the spectral index and the relative flux ratio, can be used to model the central star while reconstructing the image of the extended structure separately. Methods. We present a new method, called SPARCO (semi-parametric algorithm for the image reconstruction of chromatic objects), which describes the spectral characteristics of both the central object and the extended structure to consider them properly when reconstructing the image of the surrounding environment. We adapted two image-reconstruction codes (Macim, Squeeze, and MiRA) to implement this new prescription. SPARCO is applied using Macim, Squeeze and MiRA on a young stellar object model and also on literature data on HR5999 in the near-infrared with the VLTI. This method paves the way to improved aperture-synthesis imaging of several young stellar objects with existing datasets. More generally, the approach can be used on astrophysical sources with similar features such as active galactic nuclei, planetary nebulae, and asymptotic giant branch stars.Comment: 11 pages, 11 figures, accepted in A&

Boxicity and separation dimension

A family $\mathcal{F}$ of permutations of the vertices of a hypergraph $H$ is called 'pairwise suitable' for $H$ if, for every pair of disjoint edges in $H$, there exists a permutation in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for $H$ is called the 'separation dimension' of $H$ and is denoted by $\pi(H)$. Equivalently, $\pi(H)$ is the smallest natural number $k$ so that the vertices of $H$ can be embedded in $\mathbb{R}^k$ such that any two disjoint edges of $H$ can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph $H$ is equal to the 'boxicity' of the line graph of $H$. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to WG-2014. Some results proved in this paper are also present in arXiv:1212.6756. arXiv admin note: substantial text overlap with arXiv:1212.675