On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

Golumbic, Lipshteyn and Stern \cite{Golumbic-epg} proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer $k$, $B_k$-EPG graphs are defined as EPG graphs admitting a model in which each path has at most $k$ bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a $B_4$-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that every circular-arc graph is $B_3$-EPG, and that there exist circular-arc graphs which are not $B_2$-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in a rectangle of the grid), we obtain EPR (edge intersection of path in a rectangle) representations. We may define $B_k$-EPR graphs, $k\geq 0$, the same way as $B_k$-EPG graphs. Circular-arc graphs are clearly $B_4$-EPR graphs and we will show that there exist circular-arc graphs that are not $B_3$-EPR graphs. We also show that normal circular-arc graphs are $B_2$-EPR graphs and that there exist normal circular-arc graphs that are not $B_1$-EPR graphs. Finally, we characterize $B_1$-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs

A giant impact as the likely origin of different twins in the Kepler-107 exoplanet system

Measures of exoplanet bulk densities indicate that small exoplanets with radius less than 3 Earth radii (R⊕) range from low-density sub-Neptunes containing volatile elements1 to higher-density rocky planets with Earth-like2 or iron-rich3 (Mercury-like) compositions. Such astonishing diversity in observed small exoplanet compositions may be the product of different initial conditions of the planet-formation process or different evolutionary paths that altered the planetary properties after formation4. Planet evolution may be especially affected by either photoevaporative mass loss induced by high stellar X-ray and extreme ultraviolet (XUV) flux5 or giant impacts6. Although there is some evidence for the former7,8, there are no unambiguous findings so far about the occurrence of giant impacts in an exoplanet system. Here, we characterize the two innermost planets of the compact and near-resonant system Kepler-107 (ref. 9). We show that they have nearly identical radii (about 1.5–1.6R⊕), but the outer planet Kepler-107 c is more than twice as dense (about 12.6 g cm–3) as the innermost Kepler-107 b (about 5.3 g cm−3). In consequence, Kepler-107 c must have a larger iron core fraction than Kepler-107 b. This imbalance cannot be explained by the stellar XUV irradiation, which would conversely make the more-irradiated and less-massive planet Kepler-107 b denser than Kepler-107 c. Instead, the dissimilar densities are consistent with a giant impact event on Kepler-107 c that would have stripped off part of its silicate mantle. This hypothesis is supported by theoretical predictions from collisional mantle stripping10, which match the mass and radius of Kepler-107 c

Exoplanet atmospheres with GIANO II. Detection of molecular absorption in the dayside spectrum of HD 102195b

The study of exoplanetary atmospheres is key to understand the differences between their physical, chemical and dynamical processes. Up to now, the bulk of atmospheric characterization analysis has been conducted on transiting planets. On some sufficiently bright targets, high-resolution spectroscopy (HRS) has also been successfully tested for non-transiting planets. We study the dayside of the non-transiting planet HD 102195b using the GIANO spectrograph mounted at TNG, demonstrating the feasibility of atmospheric characterization measurements and molecular detection for non-transiting planets with the HRS technique using 4-m class telescopes. The Doppler-shifted planetary signal changes on the order of many km/s during the observations, in contrast with the telluric absorption which is stationary in wavelength, allowing us to remove the contamination from telluric lines while preserving the features of the planetary spectrum. The emission signal from HD 102195b's atmosphere is then extracted by cross-correlating the residual spectra with atmospheric models. We detect molecular absorption from water vapor at 4.4$\sigma$ level. We also find convincing evidence for the presence of methane, which is detected at the 4.1$\sigma$ level. The two molecules are detected with a combined significance of 5.3$\sigma$, at a semi-amplitude of the planet radial velocity $K_P=128\pm 6$ km/s. We estimate a planet true mass of $M_P=0.46\pm 0.03~M_J$ and orbital inclination between 72.5 and 84.79$^{\circ}$ (1$\sigma$). Our analysis indicates a non-inverted atmosphere for HD 102195b, as expected given the relatively low temperature of the planet, inefficient to keep TiO/VO in gas phase. Moreover, a comparison with theoretical expectations and chemical model predictions corroborates our methane detection and suggests that the detected $CH_4$ and $H_2O$ signatures could be consistent with a low C/O ratio.Comment: 12 pages, 12 figures, accepted for publication in A&