Evaluating Animation Parameters for Morphing Edge Drawings

Partial edge drawings (PED) of graphs avoid edge crossings by subdividing each edge into three parts and representing only its stubs, i.e., the parts incident to the end-nodes. The morphing edge drawing model (MED) extends the PED drawing style by animations that smoothly morph each edge between its representation as stubs and the one as a fully drawn segment while avoiding new crossings. Participants of a previous study on MED (Misue and Akasaka, GD19) reported eye straining caused by the animation. We conducted a user study to evaluate how this effect is influenced by varying animation speed and animation dynamic by considering an easing technique that is commonly used in web design. Our results provide indications that the easing technique may help users in executing topology-based tasks accurately. The participants also expressed appreciation for the easing and a preference for a slow animation speed.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Weakly and Strongly Fan-Planar Graphs

We study two notions of fan-planarity introduced by (Cheong et al., GD22), called weak and strong fan-planarity that separate two non-equivalent definitions of fan-planarity in the literature. We prove that not every weakly fan-planar graph is strongly fan-planar, while the density upper bound for both families is the same.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Drawing Graphs with Circular Arcs and Right-Angle Crossings

In a RAC drawing of a graph, vertices are represented by points in the plane, adjacent vertices are connected by line segments, and crossings must form right angles. Graphs that admit such drawings are RAC graphs. RAC graphs are beyond-planar graphs and have been studied extensively. In particular, it is known that a RAC graph with n vertices has at most 4n - 10 edges. We introduce a superclass of RAC graphs, which we call arc-RAC graphs. A graph is arc-RAC if it admits a drawing where edges are represented by circular arcs and crossings form right angles. We provide a Tur\'an-type result showing that an arc-RAC graph with n vertices has at most 14n - 12 edges and that there are n-vertex arc-RAC graphs with 4.5n - o(n) edges

Drawing Shortest Paths in Geodetic Graphs

Motivated by the fact that in a space where shortest paths are unique, no two shortest paths meet twice, we study a question posed by Greg Bodwin: Given a geodetic graph $G$, i.e., an unweighted graph in which the shortest path between any pair of vertices is unique, is there a philogeodetic drawing of $G$, i.e., a drawing of $G$ in which the curves of any two shortest paths meet at most once? We answer this question in the negative by showing the existence of geodetic graphs that require some pair of shortest paths to cross at least four times. The bound on the number of crossings is tight for the class of graphs we construct. Furthermore, we exhibit geodetic graphs of diameter two that do not admit a philogeodetic drawing.Comment: Appears in the Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization (GD 2020

Physical Conditions in a Young, Unreddened, Low Metallicity Galaxy at High Redshift

Increasingly large samples of galaxies are now being discovered at redshifts z~5-6 and higher. Many of these objects are inferred to be young, low in mass, and relatively unreddened, but detailed analysis of their high quality spectra will not be possible until the advent of future facilities. In this paper we shed light on the physical conditions in a plausibly similar low mass galaxy by presenting the analysis of the rest-frame optical and UV spectra of Q2343-BX418, an L* galaxy at z=2.3 with a very low mass-to-light ratio and unusual properties: BX418 is young (<100 Myr), low mass (M_star ~ 10^9 Msun), low in metallicity (Z ~ 1/6 Zsun), and unreddened (E(B-V)~0.02, UV continuum slope beta=-2.1). We infer a metallicity 12+log(O/H)=7.9 +/- 0.2 from the rest-frame optical emission lines. We also determine the metallicity via the direct, electron temperature method, using the ratio O III] 1661, 1666/[O III] 5007 to determine the electron temperature and finding 12+ log(O/H)=7.8 +/- 0.1. These measurements place BX418 among the most metal-poor galaxies observed in emission at high redshift. The rest-frame UV spectrum contains strong emission from Lya (with rest-frame equivalent width 54 A), He II 1640 (both stellar and nebular), C III] 1907, 1909 and O III] 1661, 1666. The C IV/C III] ratio indicates that the source of ionization is unlikely to be an AGN. Analysis of the He II, O III] and C III] line strengths indicates a very high ionization parameter log U ~ -1, while Lya and the interstellar absorption lines indicate that outflowing gas is highly ionized over a wide range of velocities. It remains to be determined how many of BX418's unique spectral features are due to its global properties, such as low metallicity and dust extinction, and how many are indicative of a short-lived phase in the early evolution of an otherwise normal star-forming galaxy.Comment: Accepted for publication in ApJ. 28 pages, 14 figure

Multi-Wavelength View of Kiloparsec-Scale Clumps in Star-Forming Galaxies at z~2

This paper studies the properties of kiloparsec-scale clumps in star-forming galaxies at z~2 through multi-wavelength broad band photometry. A sample of 40 clumps is identified through auto-detection and visual inspection from 10 galaxies with 1.5<z<2.5 in the Hubble Ultra Deep Field, where deep and high-resolution HST/WFC3 and ACS images enable us to resolve structures of z~2 galaxies down to kpc scale in the rest-frame UV and optical bands as well as to detect clumps toward the faint end. The physical properties of clumps are measured through fitting spatially resolved seven-band (BVizYJH) spectral energy distribution to models. On average, the clumps are blue and have similar median rest-frame UV--optical color as the diffuse components of their host galaxies, but the clumps have large scatter in their colors. Although the star formation rate (SFR)--stellar mass relation of galaxies is dominated by the diffuse components, clumps emerge as regions with enhanced specific SFRs, contributing individually ~10% and together ~50% of the SFR of the host galaxies. However, the contributions of clumps to the rest-frame UV/optical luminosity and stellar mass are smaller, typically a few percent individually and ~20% together. On average, clumps are younger by 0.2 dex and denser by a factor of 8 than diffuse components. Clump properties have obvious radial variations in the sense that central clumps are redder, older, more extincted, denser, and less active on forming stars than outskirts clumps. Our results are broadly consistent with a widely held view that clumps are formed through gravitational instability in gas-rich turbulent disks and would eventually migrate toward galactic centers and coalesce into bulges. Roughly 40% of the galaxies in our sample contain a massive clump that could be identified as a proto-bulge, which seems qualitatively consistent with such a bulge-formation scenario.Comment: Accepted by ApJ. This updated version matches the in-press one. 50 pages (single column), 10 figures, 3 table