  We present an unbiased numerical density-matrix renormalization group study
of the one-dimensional Bose-Hubbard model supplemented by nearest-neighbor
Coulomb interaction and bond dimerization. It places the emphasis on the
determination of the ground-state phase diagram and shows that, besides
dimerized Mott and density-wave insulating phases, an intermediate
symmetry-protected topological Haldane insulator emerges at weak Coulomb
interactions for filling factor one, which disappears, however, when the
dimerization becomes too large. Analyzing the critical behavior of the model,
we prove that the phase boundaries of the Haldane phase to Mott insulator and
density-wave states belong to the Gaussian and Ising universality classes with
central charges $c=1$ and $c=1/2$, respectively, and merge in a tricritical
point. Interestingly we can demonstrate a direct Ising quantum phase transition
between the dimerized Mott and density-wave phases above the tricritical point.
The corresponding transition line terminates at a critical end point that
belongs to the universality class of the dilute Ising model with $c=7/10$. At
even stronger Coulomb interactions the transition becomes first order.
