On Vertex- and Empty-Ply Proximity Drawings

We initiate the study of the vertex-ply of straight-line drawings, as a relaxation of the recently introduced ply number. Consider the disks centered at each vertex with radius equal to half the length of the longest edge incident to the vertex. The vertex-ply of a drawing is determined by the vertex covered by the maximum number of disks. The main motivation for considering this relaxation is to relate the concept of ply to proximity drawings. In fact, if we interpret the set of disks as proximity regions, a drawing with vertex-ply number 1 can be seen as a weak proximity drawing, which we call empty-ply drawing. We show non-trivial relationships between the ply number and the vertex-ply number. Then, we focus on empty-ply drawings, proving some properties and studying what classes of graphs admit such drawings. Finally, we prove a lower bound on the ply and the vertex-ply of planar drawings.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Optimality program in segment and string graphs

Planar graphs are known to allow subexponential algorithms running in time $2^{O(\sqrt n)}$ or $2^{O(\sqrt n \log n)}$ for most of the paradigmatic problems, while the brute-force time $2^{\Theta(n)}$ is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time $2^{O(n^{2/3}\log^{O(1)}n)}$ on string graphs while an algorithm running in time $2^{o(n)}$ for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker ETH lower bound of $2^{o(n^{2/3})}$ which exploits the celebrated Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent Dominating Set.Comment: 19 pages, 15 figure

Correlation between the spatial distribution of circumstellar disks and massive stars in the young open cluster NGC 6611. II: Cluster members selected with Spitzer/IRAC

Context: the observations of the proplyds in the Orion Nebula Cluster, showing clear evidence of ongoing photoevaporation, have provided a clear proof about the role of the externally induced photoevaporation in the evolution of circumstellar disks. NGC 6611 is an open cluster suitable to study disk photoevaporation, thanks to its large population of massive members and of stars with disk. In a previous work, we obtained evidence of the influence of the strong UV field generated by the massive cluster members on the evolution of disks around low-mass Pre-Main Sequence members. That work was based on a multi-band BVIJHK and X-ray catalog purposely compiled to select the cluster members with and without disk. Aims: in this paper we complete the list of candidate cluster members, using data at longer wavelengths obtained with Spitzer/IRAC, and we revisit the issue of the effects of UV radiation on the evolution of disks in NGC 6611. Methods: we select the candidate members with disks of NGC 6611, in a field of view of 33'x34' centered on the cluster, using IRAC color-color diagrams and suitable reddening-free color indices. Besides, using the X-ray data to select Class III cluster members, we estimate the disks frequency vs. the intensity of the incident radiation emitted by massive members. Results: we identify 458 candidate members with circumstellar disks, among which 146 had not been revealed in our previous work. Comparing of the various color indices we used to select the cluster members with disk, we claim that they detect the excesses due to the emission of the same physical region of the disk: the inner rim at the dust sublimation radius. Our new results confirm that UV radiation from massive stars affects the evolution of nearby circumstellar disks.Comment: Accepted for publication at Astronomy & Astrophysic

The Maximum Exposure Problem

Given a set of points P and axis-aligned rectangles R in the plane, a point p in P is called exposed if it lies outside all rectangles in R. In the max-exposure problem, given an integer parameter k, we want to delete k rectangles from R so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible complexity conjectures is also hard to approximate even when rectangles in R are translates of two fixed rectangles. However, if R only consists of translates of a single rectangle, we present a polynomial-time approximation scheme. For general rectangle range space, we present a simple O(k) bicriteria approximation algorithm; that is by deleting O(k^2) rectangles, we can expose at least Omega(1/k) of the optimal number of points

Protoplanetary Disks in the Orion Nebula Cluster: Gas Disk Morphologies and Kinematics as seen with ALMA

We present Atacama Large Millimeter Array CO(3$-$2) and HCO$^+$(4$-$3) observations covering the central $1\rlap{.}'5$$\times$$1\rlap{.}'5$ region of the Orion Nebula Cluster (ONC). The unprecedented level of sensitivity ($\sim$0.1 mJy beam$^{-1}$) and angular resolution ($\sim$$0\rlap{.}''09 \approx 35$ AU) of these line observations enable us to search for gas-disk detections towards the known positions of submillimeter-detected dust disks in this region. We detect 23 disks in gas: 17 in CO(3$-$2), 17 in HCO$^+$(4$-$3), and 11 in both lines. Depending on where the sources are located in the ONC, we see the line detections in emission, in absorption against the warm background, or in both emission and absorption. We spectrally resolve the gas with $0.5$ km s$^{-1}$ channels, and find that the kinematics of most sources are consistent with Keplerian rotation. We measure the distribution of gas-disk sizes and find typical radii of $\sim$50-200 AU. As such, gas disks in the ONC are compact in comparison with the gas disks seen in low-density star-forming regions. Gas sizes are universally larger than the dust sizes. However, the gas and dust sizes are not strongly correlated. We find a positive correlation between gas size and distance from the massive star $\theta^1$ Ori C, indicating that disks in the ONC are influenced by photoionization. Finally, we use the observed kinematics of the detected gas lines to model Keplerian rotation and infer the masses of the central pre-main-sequence stars. Our dynamically-derived stellar masses are not consistent with the spectroscopically-derived masses, and we discuss possible reasons for this discrepancy.Comment: 42 pages, 31 figure