Search

We define a two-player combinatorial game in which players take alternate turns; each turn consists on deleting a vertex of a graph, together with all the edges containing such vertex. If any vertex became isolated by a player's move then it would also be deleted. A player wins the game when the other player has no moves available. We study this game under various viewpoints: by finding specific strategies for certain families of graphs, through using properties of a graph's automorphism group, by writing a program to look at Sprague-Grundy numbers, and by studying the game when played on random graphs. When analyzing Grim played on paths, using the Sprague-Grundy function, we find a connection to a standing open question about Octal games., Comment: 13 pages, 0 figures

We present the results of HST/WFPC2 medium and narrow band imaging and VLA and MERLIN2 radio imaging of three powerful radio galaxies: 3C 171, 3C 277.3, and PKS 2250-41. We obtained images of the rest frame [OIII]$\lambda$5007 and [OII]$\lambda$3727 line emission using the Linear Ramp Filters on WFPC2. The correlations between the emission line morphology and the [OIII]/[OII] line ratios with the radio emission seen in ground based observations are clarified by the HST imaging. We confirm that the radio lobes and hot-spots are preferentially associated with lower ionization gas. 3C 171 exhibits high surface brightness emission line gas mainly along the radio source axis. The lowest ionization gas is seen at the Eastern hot spot. In 3C 277.3 there is bright high ionization gas (and continuum) offset just to the east of the radio knot K1. Our observations are consistent with previous work suggesting that this emission is produced by precursor gas ionized by the shock being driven into the cloud by the deflected radio jet. In PKS 2250-41 we resolve the emission line arc which wraps around the outer rim of the western lobe. The lower ionization [OII] emission is nested just interior to the higher ionization [OIII] emission suggesting that we have resolved the cooling region behind the bow shock. We also detect possible continuum emission from the secondary hot-spot. Thus, our observations support the hypothesis that in these sources, the interaction between the expanding radio source and the ambient gas strongly influences the morphology, kinematics, and ionization of the gas., Comment: 27 pages, includes 12 figures and 6 tables, accepted for publication in The Astronomical Journal