Neighborhood Complexity and Kernelization for Nowhere Dense Classes of Graphs

We prove that whenever G is a graph from a nowhere dense graph class C, and A is a subset of vertices of G, then the number of subsets of A that are realized as intersections of A with r-neighborhoods of vertices of G is at most f(r,eps)|A|^(1+eps), where r is any positive integer, eps is any positive real, and f is a function that depends only on the class C. This yields a characterization of nowhere dense classes of graphs in terms of neighborhood complexity, which answers a question posed by [Reidl et al., CoRR, 2016]. As an algorithmic application of the above result, we show that for every fixed integer r, the parameterized Distance-r Dominating Set problem admits an almost linear kernel on any nowhere dense graph class. This proves a conjecture posed by [Drange et al., STACS 2016], and shows that the limit of parameterized tractability of Distance-r Dominating Set on subgraph-closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness

Gas morphology and energetics at the surface of PDRs: new insights with Herschel observations of NGC 7023

We investigate the physics and chemistry of the gas and dust in dense photon-dominated regions (PDRs), along with their dependence on the illuminating UV field. Using Herschel-HIFI observations, we study the gas energetics in NGC 7023 in relation to the morphology of this nebula. NGC 7023 is the prototype of a PDR illuminated by a B2V star and is one of the key targets of Herschel. Our approach consists in determining the energetics of the region by combining the information carried by the mid-IR spectrum (extinction by classical grains, emission from very small dust particles) with that of the main gas coolant lines. In this letter, we discuss more specifically the intensity and line profile of the 158 micron (1901 GHz) [CII] line measured by HIFI and provide information on the emitting gas. We show that both the [CII] emission and the mid-IR emission from polycyclic aromatic hydrocarbons (PAHs) arise from the regions located in the transition zone between atomic and molecular gas. Using the Meudon PDR code and a simple transfer model, we find good agreement between the calculated and observed [CII] intensities. HIFI observations of NGC 7023 provide the opportunity to constrain the energetics at the surface of PDRs. Future work will include analysis of the main coolant line [OI] and use of a new PDR model that includes PAH-related species.Comment: Accepted for publication in Astronomy and Astrophysics Letters (Herschel HIFI special issue), 5 pages, 5 figure

The origin of the [C II] emission in the S140 PDRs - new insights from HIFI

Using Herschel's HIFI instrument we have observed [C II] along a cut through S140 and high-J transitions of CO and HCO+ at two positions on the cut, corresponding to the externally irradiated ionization front and the embedded massive star forming core IRS1. The HIFI data were combined with available ground-based observations and modeled using the KOSMA-tau model for photon dominated regions. Here we derive the physical conditions in S140 and in particular the origin of [C II] emission around IRS1. We identify three distinct regions of [C II] emission from the cut, one close to the embedded source IRS1, one associated with the ionization front and one further into the cloud. The line emission can be understood in terms of a clumpy model of photon-dominated regions. At the position of IRS1, we identify at least two distinct components contributing to the [C II] emission, one of them a small, hot component, which can possibly be identified with the irradiated outflow walls. This is consistent with the fact that the [C II] peak at IRS1 coincides with shocked H2 emission at the edges of the outflow cavity. We note that previously available observations of IRS1 can be well reproduced by a single-component KOSMA-tau model. Thus it is HIFI's unprecedented spatial and spectral resolution, as well as its sensitivity which has allowed us to uncover an additional hot gas component in the S140 region.Comment: accepted for publication in Astronomy and Astrophysics (HIFI special issue

13CI in high-mass star-forming clouds

We report measurements of the 12C/13C abundance ratio in the three galactic regions G 333.0-0.4, NGC 6334 A and G 351.6-1.3 from observations of the 12CI 3P2-3P1 transition and the hyperfine components of the corresponding 13CI transition near 809 GHz. These transitions were observed simultaneously with the CO 7-6 line emission at 806 GHz with the AST/RO telescope located at the South Pole. From a simultaneous fit to the 12CI 3P2-3P1 transition and the HF components of the corresponding 13CI transition and an independent estimate of an upper limit to the optical depth of the 12CI emission we determine intrinsic 12CI/13CI column density ratios of 23+-1 for G 333.0-0.4, 56+-14 for NGC 6334 A and 69+-12 for G 351.6-1.3. As the regions observed are photon dominated, we argue that the apparent enhancement in the abundance of 13C towards G 333.0-0.4 may be due to strong isotope-selective photodissociation of 13CO, outweighing the effects of chemical isotopic fractionation as suggested by models of PDRs. Towards NGC 6334 A and G 351.6-1.3 these effects appear to be balanced, similar to the situation for the Orion Bar region observed by Keene et al. (1998).Comment: 4 pages, 2 figures, submitted to A&A Letter