  We show that facilitated spin mixtures with a tunable facilitation reproduce,
on a Bethe lattice, the simplest higher-order singularity scenario predicted by
the mode-coupling theory (MCT) of liquid-glass transition. Depending on the
facilitation strength, they yield either a hybrid glass transition or a
continuous one, with no underlying thermodynamic singularity. Similar results
are obtained for facilitated spin models on a diluted Bethe lattice. The
mechanism of dynamical arrest in these systems can be interpreted in terms of
bootstrap and standard percolation and corresponds to a crossover from a
compact to a fractal structure of the incipient spanning cluster of frozen
spins. Theoretical and numerical simulation results are fully consistent with
MCT predictions.
