67 research outputs found

    Domination parameters with number 2: interrelations and algorithmic consequences

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    In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak 22-domination number, γw2(G)\gamma_{w2}(G), the 22-domination number, γ2(G)\gamma_2(G), the {2}\{2\}-domination number, γ{2}(G)\gamma_{\{2\}}(G), the double domination number, γ×2(G)\gamma_{\times 2}(G), the total {2}\{2\}-domination number, γt{2}(G)\gamma_{t\{2\}}(G), and the total double domination number, γt×2(G)\gamma_{t\times 2}(G), where GG is a graph in which a corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, γR(G)\gamma_R(G), and two classical parameters, the domination number, γ(G)\gamma(G), and the total domination number, γt(G)\gamma_t(G), we consider 13 domination invariants in graphs GG. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain some complexity results for the studied invariants, in particular regarding the existence of approximation algorithms and inapproximability bounds.Comment: 45 pages, 4 tables, 7 figure

    Search for massive resonances decaying in to WW,WZ or ZZ bosons in proton-proton collisions at root s=13 TeV

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    Evidence for the 125 GeV Higgs boson decaying to a pair of tau leptons

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    Description and performance of track and primary-vertex reconstruction with the CMS tracker

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    A description is provided of the software algorithms developed for the CMS tracker both for reconstructing charged-particle trajectories in proton-proton interactions and for using the resulting tracks to estimate the positions of the LHC luminous region and individual primary-interaction vertices. Despite the very hostile environment at the LHC, the performance obtained with these algorithms is found to be excellent. For tbar t events under typical 2011 pileup conditions, the average track-reconstruction efficiency for promptly-produced charged particles with transverse momenta of pT > 0.9GeV is 94% for pseudorapidities of |η| < 0.9 and 85% for 0.9 < |η| < 2.5. The inefficiency is caused mainly by hadrons that undergo nuclear interactions in the tracker material. For isolated muons, the corresponding efficiencies are essentially 100%. For isolated muons of pT = 100GeV emitted at |η| < 1.4, the resolutions are approximately 2.8% in pT, and respectively, 10μm and 30μm in the transverse and longitudinal impact parameters. The position resolution achieved for reconstructed primary vertices that correspond to interesting pp collisions is 10–12μm in each of the three spatial dimensions. The tracking and vertexing software is fast and flexible, and easily adaptable to other functions, such as fast tracking for the trigger, or dedicated tracking for electrons that takes into account bremsstrahlung

    Measurement of prompt and nonprompt J/psi production in pp and pPb collisions at root s(NN)=5.02 TeV

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