Abstract: We show three linear time algorithms for constructing planar straight-line grid drawings of outerplanar graphs. The first and the second algorithm are for balanced outerplanar graphs. Both require linear area. The drawings produced by the first algorithm are not outerplanar while those produced by the second algorithm are. On the other hand, the first algorithm constructs drawings with better angular resolution. The third algorithm constructs outerplanar drawings of general outerplanar graphs with O(n 1.48 ) area. Further, we study the interplay between the area requirements of the drawings of an outerplanar graph and the area requirements of a special class of drawings of its dual tree.

Abstract: We consider drawings of trees in which all edges incident to leaves can be extended to infinite rays without crossing, partitioning the plane into infinite convex polygons Among all such drawings we seek the one maximizing the angular resolution of the drawing We find linear time algorithms for solving this problem, both for plane trees and for trees without a fixed embedding In any such drawing, the edge lengths may be set independently of the angles, without crossing; we describe multiple strategies for setting these lengths

Abstract: In this paper, we present an algorithm, called Bubble Tree, for the drawing of general rooted trees A large variety of algorithms already exists in this field However, the goal of this algorithm is to obtain a better drawing which makes a trade off between the angular resolution and the length of the edges We show that the Bubble Tree drawing algorithm provides a planar drawing with at most one bend per edge in linear running time

Abstract: Context.High-resolution, high-sensitivity multifrequency radio images of supernova remnants (SNRs) are essential in advancing the understanding of both the global SNR dynamics and particle acceleration mechanisms. Aims.In this paper we report on a new study of the SNR Puppis A based on VLA observations at 1425 MHz; the improvement represents a factor of two in angular resolution and almost ten times in sensitivity compared to the best previous image of Puppis A. This new image is used to compare with re-processed 327 MHz data and ROSAT and Chandra images to investigate morphological and spectral characteristics. Methods. The observations were carried out with the VLA in the DnC and CnB configurations in 2004. After combining with single-dish data from Parkes, an angular resolution of 34$^{\prime\prime} \times 16^{\prime\prime}$ and an rms noise of 0.5 mJy beam-1, were achieved. Archival VLA data at 327 MHz were also processed. The spectral index distribution was then determined by a direct comparison of the homogenized data at both 327 and 1425 MHz. In addition, to identify different spectral components, tomographic spectral analysis was performed. Results.The new 1425 MHz radio image reveals a highly structured border encircling a diffuse, featureless interior. In particular, the northern half of Puppis A displays a complex structure along the periphery, consisting of short arcs resembling “wave-like” features. These are oriented essentially perpendicular to the shock front on the NE side, but are tangential to the shock on the NW side. A remarkable correspondence between such “wave-like” features and spectral changes is observed. On the other hand, the brightest radio features (located to the E of the SNR and also detected in X-rays) have no counterpart in the spectral index distribution. Based on a uniform compilation of integrated flux densities between 19 and 8400 MHz, a global spectral index $\alpha = -0.52 \pm 0.03$ (${\rm S} \propto u^\alpha$) has been determined. The new 1425 MHz image of Puppis A was compared with the ROSAT X-ray image of the entire SNR and with the resolved arcsec Chandra image of the bright E region. There is good overall agreement between the radio and high resolution X-ray images. At the arcsec level, however, the agreement of the radio and X-ray images is less striking. A noticeable similarity is found between Puppis A, with its E and W extensions, and analogous morphological features observed in the SNR W50. This suggests that Puppis A could be another case of a SNR shaped by the action of energetic jets.

Abstract: A mosaic of six, deep, CCD images in the light of the [O iii]5007 Å nebular emission line has been obtained with the 1.3-m Skinakas (Crete) telescope of the filamentary nebulosity surrounding P Cygni. The [O iii]5007 Å line discriminates against confusing galactic Hii regions along the same sight-lines and the new mosaic did not include the 4.8 mag. central star; a source of artifacts in the previous lower angular resolution observations. New giant “lobes” and “shells” are found to be clustered around P Cygni which must be the relics of historic eruptions between 2400 and up to ≈105 yr ago.