Micro-Arcsecond Radio Astrometry

Astrometry provides the foundation for astrophysics. Accurate positions are required for the association of sources detected at different times or wavelengths, and distances are essential to estimate the size, luminosity, mass, and ages of most objects. Very Long Baseline Interferometry at radio wavelengths, with diffraction-limited imaging at sub-milliarcsec resolution, has long held the promise of micro-arcsecond astrometry. However, only in the past decade has this been routinely achieved. Currently, parallaxes for sources across the Milky Way are being measured with ~10 uas accuracy and proper motions of galaxies are being determined with accuracies of ~1 uas/y. The astrophysical applications of these measurements cover many fields, including star formation, evolved stars, stellar and super-massive black holes, Galactic structure, the history and fate of the Local Group, the Hubble constant, and tests of general relativity. This review summarizes the methods used and the astrophysical applications of micro-arcsecond radio astrometry.Comment: To appear in Annual Reviews of Astronomy and Astrophysics (2014

Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs

A path in an edge-colored graph $G$ is rainbow if no two edges of it are colored the same. The graph $G$ is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph $G$ is strongly rainbow-connected. The minimum number of colors needed to make $G$ rainbow-connected is known as the rainbow connection number of $G$, and is denoted by $\text{rc}(G)$. Similarly, the minimum number of colors needed to make $G$ strongly rainbow-connected is known as the strong rainbow connection number of $G$, and is denoted by $\text{src}(G)$. We prove that for every $k \geq 3$, deciding whether $\text{src}(G) \leq k$ is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an $n$-vertex split graph with a factor of $n^{1/2-\epsilon}$ for any $\epsilon > 0$ unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.Comment: 13 pages, 3 figure

Geodetic Graphs and Convexity.

A graph is geodetic if each two vertices are joined by a unique shortest path. The problem of characterizing such graphs was posed by Ore in 1962; although the geodetic graphs of diameter two have been described and classified by Stemple and Kantor, little is known of the structure of geodetic graphs in general. In this work, geodetic graphs are studied in the context of convexity in graphs: for a suitable family (PI) of paths in a graph G, an induced subgraph H of G is defined to be (PI)-convex if the vertex-set of H includes all vertices of G lying on paths in (PI) joining two vertices of H. Then G is (PI)-geodetic if each (PI)-convex hull of two vertices is a path. For the family (GAMMA) of geodesics (shortest paths) in G, the (GAMMA)-geodetic graphs are exactly the geodetic graphs of the original definition. For various families (PI), the (PI)-geodetic graphs are characterized. The central results concern the family (UPSILON) of chordless paths of length no greater than the diameter; the (UPSILON)-geodetic graphs are called ultrageodetic. For graphs of diameter one or two, the ultrageodetic graphs are exactly the geodetic graphs. A geometry (P,L,F) consists of an arbitrary set P, an arbitrary set L, and a set F (L-HOOK EQ) P x L. The point-flag graph of a geometry is defined here to be the graph with vertex-set P (UNION) F whose edges are the pairs {p,(p,1)} and {(p,1),(q,1)} with p,q (ELEM) P, 1 (ELEM) L, and (p,1),(q,1) (ELEM) F. With the aid of the Feit-Higman theorem on the nonexistence of generalized polygons and the collected results of Fuglister, Damerell-Georgiacodis, and Damerell on the nonexistence of Moore geometries, it is shown that two-connected ultrageodetic graphs of diameter greater than two are precisely the graphs obtained via the subdivision, with a constant number of new vertices, either of all of the edges incident with a single vertex in a complete graph, or of all edges of the form {p,(p,1)} in the point-flag graph of a finite projective plane