265 research outputs found

    A polynomial bound for untangling geometric planar graphs

    Get PDF
    To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n^\epsilon vertices fixed. We answer this question in the affirmative with \epsilon=1/4. The previous best known bound was \Omega((\log n / \log\log n)^{1/2}). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least (n/3)^{1/2} vertices fixed, while the best upper bound was O(n\log n)^{2/3}. We answer a question of Spillner and Wolff [arXiv:0709.0170 2007] by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than 3(n^{1/2}-1) vertices fixed. Moreover, we improve the lower bound to (n/2)^{1/2}.Comment: 14 pages, 7 figure

    rac-Carbon­yl{1-[(diphenyl­phosphino)meth­yl]ethanethiol­ato}(triphenyl­phosphine)rhodium(I)

    Get PDF
    The title compound, [Rh(C15H16PS)(C18H15P)(CO)], was synthesized from the reaction of the ligand rac-[Ph2PCH2CH(CH3)SH] with trans-[Rh(F)(CO)(PPh3)2] in a 1:1 molar ratio in toluene. The Rh atom is four-coordinated in a distorted square-planar geometry with the P—S ligand [Ph2PCH2CH(CH3)S] acting as a chelate and the PPh3 and disordered CO [site occupation factors of 0.61 (5) and 0.39 (5)] ligands completing the coordination
    • …
    corecore