Thoroughly equilibrated atomistic configurations of H-shaped polyethylene (PE) melts, obtained through a novel implementation of the double-bridging Monte Carlo algorithm [Karayiannis et al. J. Chem. Phys. 2003, 118, 2451], have been subjected to equilibrium NPT molecular dynamics (MD) simulations at T = 450 K and P = 1 atm, for times up to 4 mu s. The simulated model H-shaped systems consist of PE chains possessing a main backbone (a "crossbar") trapped between two branch points each of which is linked to two dangling arms. In our simulations, the average number of carbon atoms in the backbone ranged from 48 up to 300 corresponding to both unentangled and entangled crossbars, while the average arm length was kept relatively small (it ranged from 24 up to 50) corresponding always to unentangled arms. Our long MD simulation studies reveal the different relaxation mechanisms exhibited by an H-polymer: the rapid relaxation due to arm breathing (on the order of a few nanoseconds for the short, unentangled arms considered here, UP to C-50) and the slow branch point diffusion (on the order of a few microseconds for the size of the entangled backbones considered here, UP to C-300), which, in turn, governs the sluggish diffusive motion of the entire H-molecule. Analysis of the curves describing the time decay of the autocorrelation functions for the unit vectors directed from the branch point to the free end of the arm and from one branch point to the other reveals a number of relaxation modes, indicative of the strong cooperativity between arm and backbone relaxations in H-shaped structures. It is further observed that once Fickian diffusion is established, the mean-square displacement (msd) of the chain center-of-mass follows remarkably faithfully that of branch points. This validates from first principles the assumption of the pom-pom model [Bishko et al. Phys. Rev. Lett. 1997, 79, 2352] that all friction in an H-polymer is concentrated at the two branch points. Values of the branch point friction coefficient, zeta(b), a significant parameter entering the pom-pom model, have also been calculated. For the longest H-polymers studied, logarithmic plots of the msd of the inner crossbar segments against time are seen to exhibit the four different regimes predicted by the reptation theory of Doi-Edwards for entangled linear polymer melts, with corresponding exponents remarkably close to those of the theory. This allowed us to extract the characteristic relaxation times tau(e), tau(R), and tau(d) of the theory for all simulated systems and the value of the effective diameter, a, of the underlying tube model.